A string is a palindrome if it reads the same from the left to the right and from the right to the left. For example, the strings "kek", "abacaba", "r" and "papicipap" are palindromes, while the strings "abb" and "iq" are not.A substring $$$s[l \ldots r]$$$ ($$$1 \leq l \leq r \leq |s|$$$) of a string $$$s = s_{1}s_{2} \ldots s_{|s|}$$$ is the string $$$s_{l}s_{l + 1} \ldots s_{r}$$$.Anna does not like palindromes, so she makes her friends call her Ann. She also changes all the words she reads in a similar way. Namely, each word $$$s$$$ is changed into its longest substring that is not a palindrome. If all the substrings of $$$s$$$ are palindromes, she skips the word at all.Some time ago Ann read the word $$$s$$$. What is the word she changed it into?

The first line contains a non-empty string $$$s$$$ with length at most $$$50$$$ characters, containing lowercase English letters only.

If there is such a substring in $$$s$$$ that is not a palindrome, print the maximum length of such a substring. Otherwise print $$$0$$$. Note that there can be multiple longest substrings that are not palindromes, but their length is unique.

Note"mew" is not a palindrome, so the longest substring of it that is not a palindrome, is the string "mew" itself. Thus, the answer for the first example is $$$3$$$.The string "uffuw" is one of the longest non-palindrome substrings (of length $$$5$$$) of the string "wuffuw", so the answer for the second example is $$$5$$$.All substrings of the string "qqqqqqqq" consist of equal characters so they are palindromes. This way, there are no non-palindrome substrings. Thus, the answer for the third example is $$$0$$$.


Consider a billiard table of rectangular size $$$n \times m$$$ with four pockets. Let's introduce a coordinate system with the origin at the lower left corner (see the picture).   There is one ball at the point $$$(x, y)$$$ currently. Max comes to the table and strikes the ball. The ball starts moving along a line that is parallel to one of the axes or that makes a $$$45^{\circ}$$$ angle with them. We will assume that:   the angles between the directions of the ball before and after a collision with a side are equal,  the ball moves indefinitely long, it only stops when it falls into a pocket,  the ball can be considered as a point, it falls into a pocket if and only if its coordinates coincide with one of the pockets,  initially the ball is not in a pocket. Note that the ball can move along some side, in this case the ball will just fall into the pocket at the end of the side.Your task is to determine whether the ball will fall into a pocket eventually, and if yes, which of the four pockets it will be.

The only line contains $$$6$$$ integers $$$n$$$, $$$m$$$, $$$x$$$, $$$y$$$, $$$v_x$$$, $$$v_y$$$ ($$$1 \leq n, m \leq 10^9$$$, $$$0 \leq x \leq n$$$; $$$0 \leq y \leq m$$$; $$$-1 \leq v_x, v_y \leq 1$$$; $$$(v_x, v_y) \neq (0, 0)$$$) — the width of the table, the length of the table, the $$$x$$$-coordinate of the initial position of the ball, the $$$y$$$-coordinate of the initial position of the ball, the $$$x$$$-component of its initial speed and the $$$y$$$-component of its initial speed, respectively. It is guaranteed that the ball is not initially in a pocket.

Print the coordinates of the pocket the ball will fall into, or $$$-1$$$ if the ball will move indefinitely.

NoteThe first sample:   The second sample:   In the third sample the ball will never change its $$$y$$$ coordinate, so the ball will never fall into a pocket.


You are given a chessboard of size 1 × n. It is guaranteed that n is even. The chessboard is painted like this: "BWBW...BW".Some cells of the board are occupied by the chess pieces. Each cell contains no more than one chess piece. It is known that the total number of pieces equals to .In one step you can move one of the pieces one cell to the left or to the right. You cannot move pieces beyond the borders of the board. You also cannot move pieces to the cells that are already occupied.Your task is to place all the pieces in the cells of the same color using the minimum number of moves (all the pieces must occupy only the black cells or only the white cells after all the moves are made).

The first line of the input contains one integer n (2 ≤ n ≤ 100, n is even) — the size of the chessboard.  The second line of the input contains  integer numbers  (1 ≤ pi ≤ n) — initial positions of the pieces. It is guaranteed that all the positions are distinct.

Print one integer — the minimum number of moves you have to make to place all the pieces in the cells of the same color.

NoteIn the first example the only possible strategy is to move the piece at the position 6 to the position 5 and move the piece at the position 2 to the position 3. Notice that if you decide to place the pieces in the white cells the minimum number of moves will be 3.In the second example the possible strategy is to move  in 4 moves, then  in 3 moves,  in 2 moves and  in 1 move.


Year 2118. Androids are in mass production for decades now, and they do all the work for humans. But androids have to go to school to be able to solve creative tasks. Just like humans before.It turns out that high school struggles are not gone. If someone is not like others, he is bullied. Vasya-8800 is an economy-class android which is produced by a little-known company. His design is not perfect, his characteristics also could be better. So he is bullied by other androids.One of the popular pranks on Vasya is to force him to compare $$$x^y$$$ with $$$y^x$$$. Other androids can do it in milliseconds while Vasya's memory is too small to store such big numbers.Please help Vasya! Write a fast program to compare $$$x^y$$$ with $$$y^x$$$ for Vasya, maybe then other androids will respect him.

On the only line of input there are two integers $$$x$$$ and $$$y$$$ ($$$1 \le x, y \le 10^{9}$$$).

If $$$x^y < y^x$$$, then print '<' (without quotes). If $$$x^y > y^x$$$, then print '>' (without quotes). If $$$x^y = y^x$$$, then print '=' (without quotes).

NoteIn the first example $$$5^8 = 5 \cdot 5 \cdot 5 \cdot 5 \cdot 5 \cdot 5 \cdot 5 \cdot 5 = 390625$$$, and $$$8^5 = 8 \cdot 8 \cdot 8 \cdot 8 \cdot 8 = 32768$$$. So you should print '>'.In the second example $$$10^3 = 1000 < 3^{10} = 59049$$$.In the third example $$$6^6 = 46656 = 6^6$$$.


You are given an integer $$$n$$$ from $$$1$$$ to $$$10^{18}$$$ without leading zeroes.In one move you can swap any two adjacent digits in the given number in such a way that the resulting number will not contain leading zeroes. In other words, after each move the number you have cannot contain any leading zeroes.What is the minimum number of moves you have to make to obtain a number that is divisible by $$$25$$$? Print -1 if it is impossible to obtain a number that is divisible by $$$25$$$.

The first line contains an integer $$$n$$$ ($$$1 \le n \le 10^{18}$$$). It is guaranteed that the first (left) digit of the number $$$n$$$ is not a zero.

If it is impossible to obtain a number that is divisible by $$$25$$$, print -1. Otherwise print the minimum number of moves required to obtain such number. Note that you can swap only adjacent digits in the given number.

NoteIn the first example one of the possible sequences of moves is 5071 $$$\rightarrow$$$ 5701 $$$\rightarrow$$$ 7501 $$$\rightarrow$$$ 7510 $$$\rightarrow$$$ 7150.


When the curtains are opened, a canvas unfolds outside. Kanno marvels at all the blonde colours along the riverside — not tangerines, but blossoms instead."What a pity it's already late spring," sighs Mino with regret, "one more drizzling night and they'd be gone.""But these blends are at their best, aren't they?" Absorbed in the landscape, Kanno remains optimistic. The landscape can be expressed as a row of consecutive cells, each of which either contains a flower of colour amber or buff or canary yellow, or is empty.When a flower withers, it disappears from the cell that it originally belonged to, and it spreads petals of its colour in its two neighbouring cells (or outside the field if the cell is on the side of the landscape). In case petals fall outside the given cells, they simply become invisible.You are to help Kanno determine whether it's possible that after some (possibly none or all) flowers shed their petals, at least one of the cells contains all three colours, considering both petals and flowers. Note that flowers can wither in arbitrary order.

The first and only line of input contains a non-empty string $$$s$$$ consisting of uppercase English letters 'A', 'B', 'C' and characters '.' (dots) only ($$$\lvert s \rvert \leq 100$$$) — denoting cells containing an amber flower, a buff one, a canary yellow one, and no flowers, respectively.

Output "Yes" if it's possible that all three colours appear in some cell, and "No" otherwise. You can print each letter in any case (upper or lower).

NoteIn the first example, the buff and canary yellow flowers can leave their petals in the central cell, blending all three colours in it.In the second example, it's impossible to satisfy the requirement because there is no way that amber and buff meet in any cell.


Berland Football Cup starts really soon! Commentators from all over the world come to the event.Organizers have already built $$$n$$$ commentary boxes. $$$m$$$ regional delegations will come to the Cup. Every delegation should get the same number of the commentary boxes. If any box is left unoccupied then the delegations will be upset. So each box should be occupied by exactly one delegation.If $$$n$$$ is not divisible by $$$m$$$, it is impossible to distribute the boxes to the delegations at the moment.Organizers can build a new commentary box paying $$$a$$$ burles and demolish a commentary box paying $$$b$$$ burles. They can both build and demolish boxes arbitrary number of times (each time paying a corresponding fee). It is allowed to demolish all the existing boxes.What is the minimal amount of burles organizers should pay to satisfy all the delegations (i.e. to make the number of the boxes be divisible by $$$m$$$)?

The only line contains four integer numbers $$$n$$$, $$$m$$$, $$$a$$$ and $$$b$$$ ($$$1 \le n, m \le 10^{12}$$$, $$$1 \le a, b \le 100$$$), where $$$n$$$ is the initial number of the commentary boxes, $$$m$$$ is the number of delegations to come, $$$a$$$ is the fee to build a box and $$$b$$$ is the fee to demolish a box.

Output the minimal amount of burles organizers should pay to satisfy all the delegations (i.e. to make the number of the boxes be divisible by $$$m$$$). It is allowed that the final number of the boxes is equal to $$$0$$$.

NoteIn the first example organizers can build $$$5$$$ boxes to make the total of $$$14$$$ paying $$$3$$$ burles for the each of them.In the second example organizers can demolish $$$2$$$ boxes to make the total of $$$0$$$ paying $$$7$$$ burles for the each of them.In the third example organizers are already able to distribute all the boxes equally among the delegations, each one get $$$5$$$ boxes.


Each student eagerly awaits the day he would pass the exams successfully. Thus, Vasya was ready to celebrate, but, alas, he didn't pass it. However, many of Vasya's fellow students from the same group were more successful and celebrated after the exam.Some of them celebrated in the BugDonalds restaurant, some of them — in the BeaverKing restaurant, the most successful ones were fast enough to celebrate in both of restaurants. Students which didn't pass the exam didn't celebrate in any of those restaurants and elected to stay home to prepare for their reexamination. However, this quickly bored Vasya and he started checking celebration photos on the Kilogramm. He found out that, in total, BugDonalds was visited by $$$A$$$ students, BeaverKing — by $$$B$$$ students and $$$C$$$ students visited both restaurants. Vasya also knows that there are $$$N$$$ students in his group.Based on this info, Vasya wants to determine either if his data contradicts itself or, if it doesn't, how many students in his group didn't pass the exam. Can you help him so he won't waste his valuable preparation time?

The first line contains four integers — $$$A$$$, $$$B$$$, $$$C$$$ and $$$N$$$ ($$$0 \leq A, B, C, N \leq 100$$$).

If a distribution of $$$N$$$ students exists in which $$$A$$$ students visited BugDonalds, $$$B$$$ — BeaverKing, $$$C$$$ — both of the restaurants and at least one student is left home (it is known that Vasya didn't pass the exam and stayed at home), output one integer — amount of students (including Vasya) who did not pass the exam.  If such a distribution does not exist and Vasya made a mistake while determining the numbers $$$A$$$, $$$B$$$, $$$C$$$ or $$$N$$$ (as in samples 2 and 3), output $$$-1$$$.

NoteThe first sample describes following situation: $$$5$$$ only visited BugDonalds, $$$5$$$ students only visited BeaverKing, $$$5$$$ visited both of them and $$$5$$$ students (including Vasya) didn't pass the exam.In the second sample $$$2$$$ students only visited BugDonalds and $$$2$$$ only visited BeaverKing, but that means all $$$4$$$ students in group passed the exam which contradicts the fact that Vasya didn't pass meaning that this situation is impossible.The third sample describes a situation where $$$2$$$ students visited BugDonalds but the group has only $$$1$$$ which makes it clearly impossible.


Translator's note: in Russia's most widespread grading system, there are four grades: 5, 4, 3, 2, the higher the better, roughly corresponding to A, B, C and F respectively in American grading system.The term is coming to an end and students start thinking about their grades. Today, a professor told his students that the grades for his course would be given out automatically  — he would calculate the simple average (arithmetic mean) of all grades given out for lab works this term and round to the nearest integer. The rounding would be done in favour of the student — $$$4.5$$$ would be rounded up to $$$5$$$ (as in example 3), but $$$4.4$$$ would be rounded down to $$$4$$$.This does not bode well for Vasya who didn't think those lab works would influence anything, so he may receive a grade worse than $$$5$$$ (maybe even the dreaded $$$2$$$). However, the professor allowed him to redo some of his works of Vasya's choosing to increase his average grade. Vasya wants to redo as as few lab works as possible in order to get $$$5$$$ for the course. Of course, Vasya will get $$$5$$$ for the lab works he chooses to redo.Help Vasya — calculate the minimum amount of lab works Vasya has to redo.

The first line contains a single integer $$$n$$$ — the number of Vasya's grades ($$$1 \leq n \leq 100$$$). The second line contains $$$n$$$ integers from $$$2$$$ to $$$5$$$ — Vasya's grades for his lab works.

Output a single integer — the minimum amount of lab works that Vasya has to redo. It can be shown that Vasya can always redo enough lab works to get a $$$5$$$.

NoteIn the first sample, it is enough to redo two lab works to make two $$$4$$$s into $$$5$$$s.In the second sample, Vasya's average is already $$$4.75$$$ so he doesn't have to redo anything to get a $$$5$$$.In the second sample Vasya has to redo one lab work to get rid of one of the $$$3$$$s, that will make the average exactly $$$4.5$$$ so the final grade would be $$$5$$$.


This night wasn't easy on Vasya. His favorite team lost, and he didn't find himself victorious either — although he played perfectly, his teammates let him down every time. He had to win at least one more time, but the losestreak only grew longer and longer... It's no wonder he didn't get any sleep this night at all.In the morning, Vasya was waiting the bus to the university on the bus stop. Vasya's thoughts were hazy and so he couldn't remember the right bus' number quite right and got onto the bus with the number $$$n$$$.In the bus, Vasya thought that he could get the order of the digits in the number of the bus wrong. Futhermore, he could "see" some digits several times, but the digits he saw were definitely in the real number of the bus. For example, if Vasya saw the number 2028, it could mean that the real bus number could be 2028, 8022, 2820 or just 820. However, numbers 80, 22208, 52 definitely couldn't be the number of the bus. Also, real bus number couldn't start with the digit 0, this meaning that, for example, number 082 couldn't be the real bus number too.Given $$$n$$$, determine the total number of possible bus number variants.

The first line contains one integer $$$n$$$ ($$$1 \leq n \leq 10^{18}$$$) — the number of the bus that was seen by Vasya. It is guaranteed that this number does not start with $$$0$$$.

Output a single integer — the amount of possible variants of the real bus number.


After passing a test, Vasya got himself a box of $$$n$$$ candies. He decided to eat an equal amount of candies each morning until there are no more candies. However, Petya also noticed the box and decided to get some candies for himself.This means the process of eating candies is the following: in the beginning Vasya chooses a single integer $$$k$$$, same for all days. After that, in the morning he eats $$$k$$$ candies from the box (if there are less than $$$k$$$ candies in the box, he eats them all), then in the evening Petya eats $$$10\%$$$ of the candies remaining in the box. If there are still candies left in the box, the process repeats — next day Vasya eats $$$k$$$ candies again, and Petya — $$$10\%$$$ of the candies left in a box, and so on.If the amount of candies in the box is not divisible by $$$10$$$, Petya rounds the amount he takes from the box down. For example, if there were $$$97$$$ candies in the box, Petya would eat only $$$9$$$ of them. In particular, if there are less than $$$10$$$ candies in a box, Petya won't eat any at all.Your task is to find out the minimal amount of $$$k$$$ that can be chosen by Vasya so that he would eat at least half of the $$$n$$$ candies he initially got. Note that the number $$$k$$$ must be integer.

The first line contains a single integer $$$n$$$ ($$$1 \leq n \leq 10^{18}$$$) — the initial amount of candies in the box.

Output a single integer — the minimal amount of $$$k$$$ that would allow Vasya to eat at least half of candies he got.

NoteIn the sample, the amount of candies, with $$$k=3$$$, would change in the following way (Vasya eats first):$$$68 \to 65 \to 59 \to 56 \to 51 \to 48 \to 44 \to 41 \\ \to 37 \to 34 \to 31 \to 28 \to 26 \to 23 \to 21 \to 18 \to 17 \to 14 \\ \to 13 \to 10 \to 9 \to 6 \to 6 \to 3 \to 3 \to 0$$$.In total, Vasya would eat $$$39$$$ candies, while Petya — $$$29$$$.


Bishwock is a chess figure that consists of three squares resembling an "L-bar". This figure can be rotated by 90, 180 and 270 degrees so it can have four possible states: XX   XX   .X   X.X.   .X   XX   XX Bishwocks don't attack any squares and can even occupy on the adjacent squares as long as they don't occupy the same square. Vasya has a board with $$$2\times n$$$ squares onto which he wants to put some bishwocks. To his dismay, several squares on this board are already occupied by pawns and Vasya can't put bishwocks there. However, pawns also don't attack bishwocks and they can occupy adjacent squares peacefully.Knowing the positions of pawns on the board, help Vasya to determine the maximum amount of bishwocks he can put onto the board so that they wouldn't occupy the same squares and wouldn't occupy squares with pawns.

The input contains two nonempty strings that describe Vasya's board. Those strings contain only symbols "0" (zero) that denote the empty squares and symbols "X" (uppercase English letter) that denote the squares occupied by pawns. Strings are nonempty and are of the same length that does not exceed $$$100$$$.

Output a single integer — the maximum amount of bishwocks that can be placed onto the given board.


Nastya received a gift on New Year — a magic wardrobe. It is magic because in the end of each month the number of dresses in it doubles (i.e. the number of dresses becomes twice as large as it is in the beginning of the month).Unfortunately, right after the doubling the wardrobe eats one of the dresses (if any) with the 50% probability. It happens every month except the last one in the year. Nastya owns x dresses now, so she became interested in the expected number of dresses she will have in one year. Nastya lives in Byteland, so the year lasts for k + 1 months.Nastya is really busy, so she wants you to solve this problem. You are the programmer, after all. Also, you should find the answer modulo 109 + 7, because it is easy to see that it is always integer.

The only line contains two integers x and k (0 ≤ x, k ≤ 1018), where x is the initial number of dresses and k + 1 is the number of months in a year in Byteland.

In the only line print a single integer — the expected number of dresses Nastya will own one year later modulo 109 + 7.

NoteIn the first example a year consists on only one month, so the wardrobe does not eat dresses at all.In the second example after the first month there are 3 dresses with 50% probability and 4 dresses with 50% probability. Thus, in the end of the year there are 6 dresses with 50% probability and 8 dresses with 50% probability. This way the answer for this test is (6 + 8) / 2 = 7.


Two participants are each given a pair of distinct numbers from 1 to 9 such that there's exactly one number that is present in both pairs. They want to figure out the number that matches by using a communication channel you have access to without revealing it to you.Both participants communicated to each other a set of pairs of numbers, that includes the pair given to them. Each pair in the communicated sets comprises two different numbers.Determine if you can with certainty deduce the common number, or if you can determine with certainty that both participants know the number but you do not.

The first line contains two integers $$$n$$$ and $$$m$$$ ($$$1 \le n, m \le 12$$$) — the number of pairs the first participant communicated to the second and vice versa. The second line contains $$$n$$$ pairs of integers, each between $$$1$$$ and $$$9$$$, — pairs of numbers communicated from first participant to the second. The third line contains $$$m$$$ pairs of integers, each between $$$1$$$ and $$$9$$$, — pairs of numbers communicated from the second participant to the first. All pairs within each set are distinct (in particular, if there is a pair $$$(1,2)$$$, there will be no pair $$$(2,1)$$$ within the same set), and no pair contains the same number twice. It is guaranteed that the two sets do not contradict the statements, in other words, there is pair from the first set and a pair from the second set that share exactly one number.

If you can deduce the shared number with certainty, print that number. If you can with certainty deduce that both participants know the shared number, but you do not know it, print $$$0$$$. Otherwise print $$$-1$$$.

NoteIn the first example the first participant communicated pairs $$$(1,2)$$$ and $$$(3,4)$$$, and the second communicated $$$(1,5)$$$, $$$(3,4)$$$. Since we know that the actual pairs they received share exactly one number, it can't be that they both have $$$(3,4)$$$. Thus, the first participant has $$$(1,2)$$$ and the second has $$$(1,5)$$$, and at this point you already know the shared number is $$$1$$$.In the second example either the first participant has $$$(1,2)$$$ and the second has $$$(1,5)$$$, or the first has $$$(3,4)$$$ and the second has $$$(6,4)$$$. In the first case both of them know the shared number is $$$1$$$, in the second case both of them know the shared number is $$$4$$$. You don't have enough information to tell $$$1$$$ and $$$4$$$ apart.In the third case if the first participant was given $$$(1,2)$$$, they don't know what the shared number is, since from their perspective the second participant might have been given either $$$(1,3)$$$, in which case the shared number is $$$1$$$, or $$$(2,3)$$$, in which case the shared number is $$$2$$$. While the second participant does know the number with certainty, neither you nor the first participant do, so the output is $$$-1$$$.


You are given two squares, one with sides parallel to the coordinate axes, and another one with sides at 45 degrees to the coordinate axes. Find whether the two squares intersect.The interior of the square is considered to be part of the square, i.e. if one square is completely inside another, they intersect. If the two squares only share one common point, they are also considered to intersect.

The input data consists of two lines, one for each square, both containing 4 pairs of integers. Each pair represents coordinates of one vertex of the square. Coordinates within each line are either in clockwise or counterclockwise order. The first line contains the coordinates of the square with sides parallel to the coordinate axes, the second line contains the coordinates of the square at 45 degrees. All the values are integer and between $$$-100$$$ and $$$100$$$.

Print "Yes" if squares intersect, otherwise print "No". You can print each letter in any case (upper or lower).

NoteIn the first example the second square lies entirely within the first square, so they do intersect.In the second sample squares do not have any points in common.Here are images corresponding to the samples:


You are locked in a room with a door that has a keypad with 10 keys corresponding to digits from 0 to 9. To escape from the room, you need to enter a correct code. You also have a sequence of digits.Some keys on the keypad have fingerprints. You believe the correct code is the longest not necessarily contiguous subsequence of the sequence you have that only contains digits with fingerprints on the corresponding keys. Find such code.

The first line contains two integers $$$n$$$ and $$$m$$$ ($$$1 \le n, m \le 10$$$) representing the number of digits in the sequence you have and the number of keys on the keypad that have fingerprints. The next line contains $$$n$$$ distinct space-separated integers $$$x_1, x_2, \ldots, x_n$$$ ($$$0 \le x_i \le 9$$$) representing the sequence. The next line contains $$$m$$$ distinct space-separated integers $$$y_1, y_2, \ldots, y_m$$$ ($$$0 \le y_i \le 9$$$) — the keys with fingerprints.

In a single line print a space-separated sequence of integers representing the code. If the resulting sequence is empty, both printing nothing and printing a single line break is acceptable.

NoteIn the first example, the only digits with fingerprints are $$$1$$$, $$$2$$$ and $$$7$$$. All three of them appear in the sequence you know, $$$7$$$ first, then $$$1$$$ and then $$$2$$$. Therefore the output is 7 1 2. Note that the order is important, and shall be the same as the order in the original sequence.In the second example digits $$$0$$$, $$$1$$$, $$$7$$$ and $$$9$$$ have fingerprints, however only $$$0$$$ and $$$1$$$ appear in the original sequence. $$$1$$$ appears earlier, so the output is 1 0. Again, the order is important.


Allen has a LOT of money. He has $$$n$$$ dollars in the bank. For security reasons, he wants to withdraw it in cash (we will not disclose the reasons here). The denominations for dollar bills are $$$1$$$, $$$5$$$, $$$10$$$, $$$20$$$, $$$100$$$. What is the minimum number of bills Allen could receive after withdrawing his entire balance?

The first and only line of input contains a single integer $$$n$$$ ($$$1 \le n \le 10^9$$$).

Output the minimum number of bills that Allen could receive.

NoteIn the first sample case, Allen can withdraw this with a $$$100$$$ dollar bill, a $$$20$$$ dollar bill, and a $$$5$$$ dollar bill. There is no way for Allen to receive $$$125$$$ dollars in one or two bills.In the second sample case, Allen can withdraw two $$$20$$$ dollar bills and three $$$1$$$ dollar bills.In the third sample case, Allen can withdraw $$$100000000$$$ (ten million!) $$$100$$$ dollar bills.


Let's introduce a number system which is based on a roman digits. There are digits I, V, X, L which correspond to the numbers $$$1$$$, $$$5$$$, $$$10$$$ and $$$50$$$ respectively. The use of other roman digits is not allowed.Numbers in this system are written as a sequence of one or more digits. We define the value of the sequence simply as the sum of digits in it.For example, the number XXXV evaluates to $$$35$$$ and the number IXI — to $$$12$$$.Pay attention to the difference to the traditional roman system — in our system any sequence of digits is valid, moreover the order of digits doesn't matter, for example IX means $$$11$$$, not $$$9$$$.One can notice that this system is ambiguous, and some numbers can be written in many different ways. Your goal is to determine how many distinct integers can be represented by exactly $$$n$$$ roman digits I, V, X, L.

The only line of the input file contains a single integer $$$n$$$ ($$$1 \le n \le 10^9$$$) — the number of roman digits to use.

Output a single integer — the number of distinct integers which can be represented using $$$n$$$ roman digits exactly.

NoteIn the first sample there are exactly $$$4$$$ integers which can be represented — I, V, X and L.In the second sample it is possible to represent integers $$$2$$$ (II), $$$6$$$ (VI), $$$10$$$ (VV), $$$11$$$ (XI), $$$15$$$ (XV), $$$20$$$ (XX), $$$51$$$ (IL), $$$55$$$ (VL), $$$60$$$ (XL) and $$$100$$$ (LL).


On one of the planets of Solar system, in Atmosphere University, many students are fans of bingo game.It is well known that one month on this planet consists of $$$n^2$$$ days, so calendars, represented as square matrix $$$n$$$ by $$$n$$$ are extremely popular.Weather conditions are even more unusual. Due to the unique composition of the atmosphere, when interacting with sunlight, every day sky takes one of three colors: blue, green or red.To play the bingo, you need to observe the sky for one month — after each day, its cell is painted with the color of the sky in that day, that is, blue, green or red.At the end of the month, students examine the calendar. If at least one row or column contains only cells of one color, that month is called lucky.Let's call two colorings of calendar different, if at least one cell has different colors in them. It is easy to see that there are $$$3^{n \cdot n}$$$ different colorings. How much of them are lucky? Since this number can be quite large, print it modulo $$$998244353$$$.

The first and only line of input contains a single integer $$$n$$$ ($$$1 \le n \le 1000\,000$$$) — the number of rows and columns in the calendar.

Print one number — number of lucky colorings of the calendar modulo $$$998244353$$$

NoteIn the first sample any coloring is lucky, since the only column contains cells of only one color.In the second sample, there are a lot of lucky colorings, in particular, the following colorings are lucky:  While these colorings are not lucky:


There are a lot of things which could be cut — trees, paper, "the rope". In this problem you are going to cut a sequence of integers.There is a sequence of integers, which contains the equal number of even and odd numbers. Given a limited budget, you need to make maximum possible number of cuts such that each resulting segment will have the same number of odd and even integers.Cuts separate a sequence to continuous (contiguous) segments. You may think about each cut as a break between two adjacent elements in a sequence. So after cutting each element belongs to exactly one segment. Say, $$$[4, 1, 2, 3, 4, 5, 4, 4, 5, 5]$$$ $$$\to$$$ two cuts $$$\to$$$ $$$[4, 1 | 2, 3, 4, 5 | 4, 4, 5, 5]$$$. On each segment the number of even elements should be equal to the number of odd elements.The cost of the cut between $$$x$$$ and $$$y$$$ numbers is $$$|x - y|$$$ bitcoins. Find the maximum possible number of cuts that can be made while spending no more than $$$B$$$ bitcoins.

First line of the input contains an integer $$$n$$$ ($$$2 \le n \le 100$$$) and an integer $$$B$$$ ($$$1 \le B \le 100$$$) — the number of elements in the sequence and the number of bitcoins you have. Second line contains $$$n$$$ integers: $$$a_1$$$, $$$a_2$$$, ..., $$$a_n$$$ ($$$1 \le a_i \le 100$$$) — elements of the sequence, which contains the equal number of even and odd numbers

Print the maximum possible number of cuts which can be made while spending no more than $$$B$$$ bitcoins.

NoteIn the first sample the optimal answer is to split sequence between $$$2$$$ and $$$5$$$. Price of this cut is equal to $$$3$$$ bitcoins.In the second sample it is not possible to make even one cut even with unlimited number of bitcoins.In the third sample the sequence should be cut between $$$2$$$ and $$$3$$$, and between $$$4$$$ and $$$5$$$. The total price of the cuts is $$$1 + 1 = 2$$$ bitcoins.


A string $$$s$$$ of length $$$n$$$ can be encrypted by the following algorithm:  iterate over all divisors of $$$n$$$ in decreasing order (i.e. from $$$n$$$ to $$$1$$$),  for each divisor $$$d$$$, reverse the substring $$$s[1 \dots d]$$$ (i.e. the substring which starts at position $$$1$$$ and ends at position $$$d$$$). For example, the above algorithm applied to the string $$$s$$$="codeforces" leads to the following changes: "codeforces" $$$\to$$$ "secrofedoc" $$$\to$$$ "orcesfedoc" $$$\to$$$ "rocesfedoc" $$$\to$$$ "rocesfedoc" (obviously, the last reverse operation doesn't change the string because $$$d=1$$$).You are given the encrypted string $$$t$$$. Your task is to decrypt this string, i.e., to find a string $$$s$$$ such that the above algorithm results in string $$$t$$$. It can be proven that this string $$$s$$$ always exists and is unique.

The first line of input consists of a single integer $$$n$$$ ($$$1 \le n \le 100$$$) — the length of the string $$$t$$$. The second line of input consists of the string $$$t$$$. The length of $$$t$$$ is $$$n$$$, and it consists only of lowercase Latin letters.

Print a string $$$s$$$ such that the above algorithm results in $$$t$$$.

NoteThe first example is described in the problem statement.


Mishka started participating in a programming contest. There are $$$n$$$ problems in the contest. Mishka's problem-solving skill is equal to $$$k$$$.Mishka arranges all problems from the contest into a list. Because of his weird principles, Mishka only solves problems from one of the ends of the list. Every time, he chooses which end (left or right) he will solve the next problem from. Thus, each problem Mishka solves is either the leftmost or the rightmost problem in the list.Mishka cannot solve a problem with difficulty greater than $$$k$$$. When Mishka solves the problem, it disappears from the list, so the length of the list decreases by $$$1$$$. Mishka stops when he is unable to solve any problem from any end of the list.How many problems can Mishka solve?

The first line of input contains two integers $$$n$$$ and $$$k$$$ ($$$1 \le n, k \le 100$$$) — the number of problems in the contest and Mishka's problem-solving skill. The second line of input contains $$$n$$$ integers $$$a_1, a_2, \dots, a_n$$$ ($$$1 \le a_i \le 100$$$), where $$$a_i$$$ is the difficulty of the $$$i$$$-th problem. The problems are given in order from the leftmost to the rightmost in the list.

Print one integer — the maximum number of problems Mishka can solve.

NoteIn the first example, Mishka can solve problems in the following order: $$$[4, 2, 3, 1, 5, 1, 6, 4] \rightarrow [2, 3, 1, 5, 1, 6, 4] \rightarrow [2, 3, 1, 5, 1, 6] \rightarrow [3, 1, 5, 1, 6] \rightarrow [1, 5, 1, 6] \rightarrow [5, 1, 6]$$$, so the number of solved problems will be equal to $$$5$$$.In the second example, Mishka can't solve any problem because the difficulties of problems from both ends are greater than $$$k$$$.In the third example, Mishka's solving skill is so amazing that he can solve all the problems.


Polycarp has $$$n$$$ coins, the value of the $$$i$$$-th coin is $$$a_i$$$. Polycarp wants to distribute all the coins between his pockets, but he cannot put two coins with the same value into the same pocket.For example, if Polycarp has got six coins represented as an array $$$a = [1, 2, 4, 3, 3, 2]$$$, he can distribute the coins into two pockets as follows: $$$[1, 2, 3], [2, 3, 4]$$$.Polycarp wants to distribute all the coins with the minimum number of used pockets. Help him to do that.

The first line of the input contains one integer $$$n$$$ ($$$1 \le n \le 100$$$) — the number of coins. The second line of the input contains $$$n$$$ integers $$$a_1, a_2, \dots, a_n$$$ ($$$1 \le a_i \le 100$$$) — values of coins.

Print only one integer — the minimum number of pockets Polycarp needs to distribute all the coins so no two coins with the same value are put into the same pocket.


Vitya has just started learning Berlanese language. It is known that Berlanese uses the Latin alphabet. Vowel letters are "a", "o", "u", "i", and "e". Other letters are consonant.In Berlanese, there has to be a vowel after every consonant, but there can be any letter after any vowel. The only exception is a consonant "n"; after this letter, there can be any letter (not only a vowel) or there can be no letter at all. For example, the words "harakiri", "yupie", "man", and "nbo" are Berlanese while the words "horse", "king", "my", and "nz" are not.Help Vitya find out if a word $$$s$$$ is Berlanese.

The first line of the input contains the string $$$s$$$ consisting of $$$|s|$$$ ($$$1\leq |s|\leq 100$$$) lowercase Latin letters.

Print "YES" (without quotes) if there is a vowel after every consonant except "n", otherwise print "NO". You can print each letter in any case (upper or lower).

NoteIn the first and second samples, a vowel goes after each consonant except "n", so the word is Berlanese.In the third sample, the consonant "c" goes after the consonant "r", and the consonant "s" stands on the end, so the word is not Berlanese.


Natasha is planning an expedition to Mars for $$$n$$$ people. One of the important tasks is to provide food for each participant.The warehouse has $$$m$$$ daily food packages. Each package has some food type $$$a_i$$$.Each participant must eat exactly one food package each day. Due to extreme loads, each participant must eat the same food type throughout the expedition. Different participants may eat different (or the same) types of food.Formally, for each participant $$$j$$$ Natasha should select his food type $$$b_j$$$ and each day $$$j$$$-th participant will eat one food package of type $$$b_j$$$. The values $$$b_j$$$ for different participants may be different.What is the maximum possible number of days the expedition can last, following the requirements above?

The first line contains two integers $$$n$$$ and $$$m$$$ ($$$1 \le n \le 100$$$, $$$1 \le m \le 100$$$) — the number of the expedition participants and the number of the daily food packages available. The second line contains sequence of integers $$$a_1, a_2, \dots, a_m$$$ ($$$1 \le a_i \le 100$$$), where $$$a_i$$$ is the type of $$$i$$$-th food package.

Print the single integer — the number of days the expedition can last. If it is not possible to plan the expedition for even one day, print 0.

NoteIn the first example, Natasha can assign type $$$1$$$ food to the first participant, the same type $$$1$$$ to the second, type $$$5$$$ to the third and type $$$2$$$ to the fourth. In this case, the expedition can last for $$$2$$$ days, since each participant can get two food packages of his food type (there will be used $$$4$$$ packages of type $$$1$$$, two packages of type $$$2$$$ and two packages of type $$$5$$$).In the second example, there are $$$100$$$ participants and only $$$1$$$ food package. In this case, the expedition can't last even $$$1$$$ day.


Natasha is going to fly to Mars. She needs to build a rocket, which consists of several stages in some order. Each of the stages is defined by a lowercase Latin letter. This way, the rocket can be described by the string — concatenation of letters, which correspond to the stages.There are $$$n$$$ stages available. The rocket must contain exactly $$$k$$$ of them. Stages in the rocket should be ordered by their weight. So, after the stage with some letter can go only stage with a letter, which is at least two positions after in the alphabet (skipping one letter in between, or even more). For example, after letter 'c' can't go letters 'a', 'b', 'c' and 'd', but can go letters 'e', 'f', ..., 'z'.For the rocket to fly as far as possible, its weight should be minimal. The weight of the rocket is equal to the sum of the weights of its stages. The weight of the stage is the number of its letter in the alphabet. For example, the stage 'a 'weighs one ton,' b 'weighs two tons, and' z' — $$$26$$$ tons.Build the rocket with the minimal weight or determine, that it is impossible to build a rocket at all. Each stage can be used at most once.

The first line of input contains two integers — $$$n$$$ and $$$k$$$ ($$$1 \le k \le n \le 50$$$) – the number of available stages and the number of stages to use in the rocket. The second line contains string $$$s$$$, which consists of exactly $$$n$$$ lowercase Latin letters. Each letter defines a new stage, which can be used to build the rocket. Each stage can be used at most once.

Print a single integer — the minimal total weight of the rocket or -1, if it is impossible to build the rocket at all.

NoteIn the first example, the following rockets satisfy the condition: "adx" (weight is $$$1+4+24=29$$$); "ady" (weight is $$$1+4+25=30$$$); "bdx" (weight is $$$2+4+24=30$$$); "bdy" (weight is $$$2+4+25=31$$$).Rocket "adx" has the minimal weight, so the answer is $$$29$$$.In the second example, target rocket is "belo". Its weight is $$$2+5+12+15=34$$$.In the third example, $$$n=k=2$$$, so the rocket must have both stages: 'a' and 'b'. This rocket doesn't satisfy the condition, because these letters are adjacent in the alphabet. Answer is -1.


There are two strings s and t, consisting only of letters a and b. You can make the following operation several times: choose a prefix of s, a prefix of t and swap them. Prefixes can be empty, also a prefix can coincide with a whole string. Your task is to find a sequence of operations after which one of the strings consists only of a letters and the other consists only of b letters. The number of operations should be minimized.

The first line contains a string s (1 ≤ |s| ≤ 2·105). The second line contains a string t (1 ≤ |t| ≤ 2·105). Here |s| and |t| denote the lengths of s and t, respectively. It is guaranteed that at least one of the strings contains at least one a letter and at least one of the strings contains at least one b letter.

The first line should contain a single integer n (0 ≤ n ≤ 5·105) — the number of operations. Each of the next n lines should contain two space-separated integers ai, bi — the lengths of prefixes of s and t to swap, respectively. If there are multiple possible solutions, you can print any of them. It's guaranteed that a solution with given constraints exists.

NoteIn the first example, you can solve the problem in two operations:  Swap the prefix of the first string with length 1 and the prefix of the second string with length 0. After this swap, you'll have strings ab and bbb.  Swap the prefix of the first string with length 1 and the prefix of the second string with length 3. After this swap, you'll have strings bbbb and a. In the second example, the strings are already appropriate, so no operations are needed.


You are given a bracket sequence $$$s$$$ (not necessarily a regular one). A bracket sequence is a string containing only characters '(' and ')'.A regular bracket sequence is a bracket sequence that can be transformed into a correct arithmetic expression by inserting characters '1' and '+' between the original characters of the sequence. For example, bracket sequences "()()" and "(())" are regular (the resulting expressions are: "(1)+(1)" and "((1+1)+1)"), and ")(", "(" and ")" are not.Your problem is to calculate the number of regular bracket sequences of length $$$2n$$$ containing the given bracket sequence $$$s$$$ as a substring (consecutive sequence of characters) modulo $$$10^9+7$$$ ($$$1000000007$$$).

The first line of the input contains one integer $$$n$$$ ($$$1 \le n \le 100$$$) — the half-length of the resulting regular bracket sequences (the resulting sequences must have length equal to $$$2n$$$). The second line of the input contains one string $$$s$$$ ($$$1 \le |s| \le 200$$$) — the string $$$s$$$ that should be a substring in each of the resulting regular bracket sequences ($$$|s|$$$ is the length of $$$s$$$).

Print only one integer — the number of regular bracket sequences containing the given bracket sequence $$$s$$$ as a substring. Since this number can be huge, print it modulo $$$10^9+7$$$ ($$$1000000007$$$).

NoteAll regular bracket sequences satisfying the conditions above for the first example:   "(((()))())";  "((()()))()";  "((()))()()";  "(()(()))()";  "()((()))()". All regular bracket sequences satisfying the conditions above for the second example:   "((()))";  "(()())";  "(())()";  "()(())". And there is no regular bracket sequences of length $$$4$$$ containing "(((" as a substring in the third example.


There is a beautiful garden of stones in Innopolis.Its most beautiful place is the $$$n$$$ piles with stones numbered from $$$1$$$ to $$$n$$$.EJOI participants have visited this place twice. When they first visited it, the number of stones in piles was $$$x_1, x_2, \ldots, x_n$$$, correspondingly. One of the participants wrote down this sequence in a notebook. They visited it again the following day, and the number of stones in piles was equal to $$$y_1, y_2, \ldots, y_n$$$. One of the participants also wrote it down in a notebook.It is well known that every member of the EJOI jury during the night either sits in the room $$$108$$$ or comes to the place with stones. Each jury member who comes there either takes one stone for himself or moves one stone from one pile to another. We can assume that there is an unlimited number of jury members. No one except the jury goes to the place with stones at night.Participants want to know whether their notes can be correct or they are sure to have made a mistake.

The first line of the input file contains a single integer $$$n$$$, the number of piles with stones in the garden ($$$1 \leq n \leq 50$$$). The second line contains $$$n$$$ integers separated by spaces $$$x_1, x_2, \ldots, x_n$$$, the number of stones in piles recorded in the notebook when the participants came to the place with stones for the first time ($$$0 \leq x_i \leq 1000$$$). The third line contains $$$n$$$ integers separated by spaces $$$y_1, y_2, \ldots, y_n$$$, the number of stones in piles recorded in the notebook when the participants came to the place with stones for the second time ($$$0 \leq y_i \leq 1000$$$).

If the records can be consistent output "Yes", otherwise output "No" (quotes for clarity).

NoteIn the first example, the following could have happened during the night: one of the jury members moved one stone from the second pile to the first pile, and the other jury member moved one stone from the fourth pile to the third pile.In the second example, the jury took stones from the second and fourth piles.It can be proved that it is impossible for the jury members to move and took stones to convert the first array into the second array.


Tanechka is shopping in the toy shop. There are exactly $$$n$$$ toys in the shop for sale, the cost of the $$$i$$$-th toy is $$$i$$$ burles. She wants to choose two toys in such a way that their total cost is $$$k$$$ burles. How many ways to do that does she have?Each toy appears in the shop exactly once. Pairs $$$(a, b)$$$ and $$$(b, a)$$$ are considered equal. Pairs $$$(a, b)$$$, where $$$a=b$$$, are not allowed.

The first line of the input contains two integers $$$n$$$, $$$k$$$ ($$$1 \le n, k \le 10^{14}$$$) — the number of toys and the expected total cost of the pair of toys.

Print the number of ways to choose the pair of toys satisfying the condition above. Print 0, if Tanechka can choose no pair of toys in such a way that their total cost is $$$k$$$ burles.

NoteIn the first example Tanechka can choose the pair of toys ($$$1, 4$$$) or the pair of toys ($$$2, 3$$$).In the second example Tanechka can choose only the pair of toys ($$$7, 8$$$).In the third example choosing any pair of toys will lead to the total cost less than $$$20$$$. So the answer is 0.In the fourth example she can choose the following pairs: $$$(1, 1000000000000)$$$, $$$(2, 999999999999)$$$, $$$(3, 999999999998)$$$, ..., $$$(500000000000, 500000000001)$$$. The number of such pairs is exactly $$$500000000000$$$.


You are given a square board, consisting of $$$n$$$ rows and $$$n$$$ columns. Each tile in it should be colored either white or black.Let's call some coloring beautiful if each pair of adjacent rows are either the same or different in every position. The same condition should be held for the columns as well.Let's call some coloring suitable if it is beautiful and there is no rectangle of the single color, consisting of at least $$$k$$$ tiles.Your task is to count the number of suitable colorings of the board of the given size.Since the answer can be very large, print it modulo $$$998244353$$$.

A single line contains two integers $$$n$$$ and $$$k$$$ ($$$1 \le n \le 500$$$, $$$1 \le k \le n^2$$$) — the number of rows and columns of the board and the maximum number of tiles inside the rectangle of the single color, respectively.

Print a single integer — the number of suitable colorings of the board of the given size modulo $$$998244353$$$.

NoteBoard of size $$$1 \times 1$$$ is either a single black tile or a single white tile. Both of them include a rectangle of a single color, consisting of $$$1$$$ tile.Here are the beautiful colorings of a board of size $$$2 \times 2$$$ that don't include rectangles of a single color, consisting of at least $$$3$$$ tiles:  The rest of beautiful colorings of a board of size $$$2 \times 2$$$ are the following:


The campus has $$$m$$$ rooms numbered from $$$0$$$ to $$$m - 1$$$. Also the $$$x$$$-mouse lives in the campus. The $$$x$$$-mouse is not just a mouse: each second $$$x$$$-mouse moves from room $$$i$$$ to the room $$$i \cdot x \mod{m}$$$ (in fact, it teleports from one room to another since it doesn't visit any intermediate room). Starting position of the $$$x$$$-mouse is unknown.You are responsible to catch the $$$x$$$-mouse in the campus, so you are guessing about minimum possible number of traps (one trap in one room) you need to place. You are sure that if the $$$x$$$-mouse enters a trapped room, it immediately gets caught.And the only observation you made is $$$\text{GCD} (x, m) = 1$$$.

The only line contains two integers $$$m$$$ and $$$x$$$ ($$$2 \le m \le 10^{14}$$$, $$$1 \le x < m$$$, $$$\text{GCD} (x, m) = 1$$$) — the number of rooms and the parameter of $$$x$$$-mouse.

Print the only integer — minimum number of traps you need to install to catch the $$$x$$$-mouse.

NoteIn the first example you can, for example, put traps in rooms $$$0$$$, $$$2$$$, $$$3$$$. If the $$$x$$$-mouse starts in one of this rooms it will be caught immediately. If $$$x$$$-mouse starts in the $$$1$$$-st rooms then it will move to the room $$$3$$$, where it will be caught.In the second example you can put one trap in room $$$0$$$ and one trap in any other room since $$$x$$$-mouse will visit all rooms $$$1..m-1$$$ if it will start in any of these rooms.


There is an infinite board of square tiles. Initially all tiles are white.Vova has a red marker and a blue marker. Red marker can color $$$a$$$ tiles. Blue marker can color $$$b$$$ tiles. If some tile isn't white then you can't use marker of any color on it. Each marker must be drained completely, so at the end there should be exactly $$$a$$$ red tiles and exactly $$$b$$$ blue tiles across the board.Vova wants to color such a set of tiles that:  they would form a rectangle, consisting of exactly $$$a+b$$$ colored tiles;  all tiles of at least one color would also form a rectangle. Here are some examples of correct colorings:  Here are some examples of incorrect colorings:  Among all correct colorings Vova wants to choose the one with the minimal perimeter. What is the minimal perimeter Vova can obtain?It is guaranteed that there exists at least one correct coloring.

A single line contains two integers $$$a$$$ and $$$b$$$ ($$$1 \le a, b \le 10^{14}$$$) — the number of tiles red marker should color and the number of tiles blue marker should color, respectively.

Print a single integer — the minimal perimeter of a colored rectangle Vova can obtain by coloring exactly $$$a$$$ tiles red and exactly $$$b$$$ tiles blue. It is guaranteed that there exists at least one correct coloring.

NoteThe first four examples correspond to the first picture of the statement.Note that for there exist multiple correct colorings for all of the examples.In the first example you can also make a rectangle with sides $$$1$$$ and $$$8$$$, though its perimeter will be $$$18$$$ which is greater than $$$8$$$.In the second example you can make the same resulting rectangle with sides $$$3$$$ and $$$4$$$, but red tiles will form the rectangle with sides $$$1$$$ and $$$3$$$ and blue tiles will form the rectangle with sides $$$3$$$ and $$$3$$$.


There is an infinite line consisting of cells. There are $$$n$$$ boxes in some cells of this line. The $$$i$$$-th box stands in the cell $$$a_i$$$ and has weight $$$w_i$$$. All $$$a_i$$$ are distinct, moreover, $$$a_{i - 1} < a_i$$$ holds for all valid $$$i$$$.You would like to put together some boxes. Putting together boxes with indices in the segment $$$[l, r]$$$ means that you will move some of them in such a way that their positions will form some segment $$$[x, x + (r - l)]$$$.In one step you can move any box to a neighboring cell if it isn't occupied by another box (i.e. you can choose $$$i$$$ and change $$$a_i$$$ by $$$1$$$, all positions should remain distinct). You spend $$$w_i$$$ units of energy moving the box $$$i$$$ by one cell. You can move any box any number of times, in arbitrary order.Sometimes weights of some boxes change, so you have queries of two types:   $$$id$$$ $$$nw$$$ — weight $$$w_{id}$$$ of the box $$$id$$$ becomes $$$nw$$$.  $$$l$$$ $$$r$$$ — you should compute the minimum total energy needed to put together boxes with indices in $$$[l, r]$$$. Since the answer can be rather big, print the remainder it gives when divided by $$$1000\,000\,007 = 10^9 + 7$$$. Note that the boxes are not moved during the query, you only should compute the answer. Note that you should minimize the answer, not its remainder modulo $$$10^9 + 7$$$. So if you have two possible answers $$$2 \cdot 10^9 + 13$$$ and $$$2 \cdot 10^9 + 14$$$, you should choose the first one and print $$$10^9 + 6$$$, even though the remainder of the second answer is $$$0$$$.

The first line contains two integers $$$n$$$ and $$$q$$$ ($$$1 \le n, q \le 2 \cdot 10^5$$$) — the number of boxes and the number of queries. The second line contains $$$n$$$ integers $$$a_1, a_2, \dots a_n$$$ ($$$1 \le a_i \le 10^9$$$) — the positions of the boxes. All $$$a_i$$$ are distinct, $$$a_{i - 1} < a_i$$$ holds for all valid $$$i$$$. The third line contains $$$n$$$ integers $$$w_1, w_2, \dots w_n$$$ ($$$1 \le w_i \le 10^9$$$) — the initial weights of the boxes. Next $$$q$$$ lines describe queries, one query per line. Each query is described in a single line, containing two integers $$$x$$$ and $$$y$$$. If $$$x < 0$$$, then this query is of the first type, where $$$id = -x$$$, $$$nw = y$$$ ($$$1 \le id \le n$$$, $$$1 \le nw \le 10^9$$$). If $$$x > 0$$$, then the query is of the second type, where $$$l = x$$$ and $$$r = y$$$ ($$$1 \le l_j \le r_j \le n$$$). $$$x$$$ can not be equal to $$$0$$$.

For each query of the second type print the answer on a separate line. Since answer can be large, print the remainder it gives when divided by $$$1000\,000\,007 = 10^9 + 7$$$.

NoteLet's go through queries of the example:   $$$1\ 1$$$ — there is only one box so we don't need to move anything.  $$$1\ 5$$$ — we can move boxes to segment $$$[4, 8]$$$: $$$1 \cdot |1 - 4| + 1 \cdot |2 - 5| + 1 \cdot |6 - 6| + 1 \cdot |7 - 7| + 2 \cdot |10 - 8| = 10$$$.  $$$1\ 3$$$ — we can move boxes to segment $$$[1, 3]$$$.  $$$3\ 5$$$ — we can move boxes to segment $$$[7, 9]$$$.  $$$-3\ 5$$$ — $$$w_3$$$ is changed from $$$1$$$ to $$$5$$$.  $$$-1\ 10$$$ — $$$w_1$$$ is changed from $$$1$$$ to $$$10$$$. The weights are now equal to $$$w = [10, 1, 5, 1, 2]$$$.  $$$1\ 4$$$ — we can move boxes to segment $$$[1, 4]$$$.  $$$2\ 5$$$ — we can move boxes to segment $$$[5, 8]$$$.


You are given a tuple generator $$$f^{(k)} = (f_1^{(k)}, f_2^{(k)}, \dots, f_n^{(k)})$$$, where $$$f_i^{(k)} = (a_i \cdot f_i^{(k - 1)} + b_i) \bmod p_i$$$ and $$$f^{(0)} = (x_1, x_2, \dots, x_n)$$$. Here $$$x \bmod y$$$ denotes the remainder of $$$x$$$ when divided by $$$y$$$. All $$$p_i$$$ are primes.One can see that with fixed sequences $$$x_i$$$, $$$y_i$$$, $$$a_i$$$ the tuples $$$f^{(k)}$$$ starting from some index will repeat tuples with smaller indices. Calculate the maximum number of different tuples (from all $$$f^{(k)}$$$ for $$$k \ge 0$$$) that can be produced by this generator, if $$$x_i$$$, $$$a_i$$$, $$$b_i$$$ are integers in the range $$$[0, p_i - 1]$$$ and can be chosen arbitrary. The answer can be large, so print the remainder it gives when divided by $$$10^9 + 7$$$

The first line contains one integer $$$n$$$ ($$$1 \le n \le 2 \cdot 10^5$$$) — the number of elements in the tuple. The second line contains $$$n$$$ space separated prime numbers — the modules $$$p_1, p_2, \ldots, p_n$$$ ($$$2 \le p_i \le 2 \cdot 10^6$$$).

Print one integer — the maximum number of different tuples modulo $$$10^9 + 7$$$.


Recently Vasya found a golden ticket — a sequence which consists of $$$n$$$ digits $$$a_1a_2\dots a_n$$$. Vasya considers a ticket to be lucky if it can be divided into two or more non-intersecting segments with equal sums. For example, ticket $$$350178$$$ is lucky since it can be divided into three segments $$$350$$$, $$$17$$$ and $$$8$$$: $$$3+5+0=1+7=8$$$. Note that each digit of sequence should belong to exactly one segment.Help Vasya! Tell him if the golden ticket he found is lucky or not.

The first line contains one integer $$$n$$$ ($$$2 \le n \le 100$$$) — the number of digits in the ticket. The second line contains $$$n$$$ digits $$$a_1 a_2 \dots a_n$$$ ($$$0 \le a_i \le 9$$$) — the golden ticket. Digits are printed without spaces.

If the golden ticket is lucky then print "YES", otherwise print "NO" (both case insensitive).

NoteIn the first example the ticket can be divided into $$$7$$$, $$$34$$$ and $$$52$$$: $$$7=3+4=5+2$$$.In the second example it is impossible to divide ticket into segments with equal sum.


You have a plate and you want to add some gilding to it. The plate is a rectangle that we split into $$$w\times h$$$ cells. There should be $$$k$$$ gilded rings, the first one should go along the edge of the plate, the second one — $$$2$$$ cells away from the edge and so on. Each ring has a width of $$$1$$$ cell. Formally, the $$$i$$$-th of these rings should consist of all bordering cells on the inner rectangle of size $$$(w - 4(i - 1))\times(h - 4(i - 1))$$$.  The picture corresponds to the third example. Your task is to compute the number of cells to be gilded.

The only line contains three integers $$$w$$$, $$$h$$$ and $$$k$$$ ($$$3 \le w, h \le 100$$$, $$$1 \le k \le \left\lfloor \frac{min(n, m) + 1}{4}\right\rfloor$$$, where $$$\lfloor x \rfloor$$$ denotes the number $$$x$$$ rounded down) — the number of rows, columns and the number of rings, respectively.

Print a single positive integer — the number of cells to be gilded.

NoteThe first example is shown on the picture below.  The second example is shown on the picture below.  The third example is shown in the problem description.


You are given two positive integers $$$a$$$ and $$$b$$$. There are two possible operations:   multiply one of the numbers by some prime $$$p$$$;  divide one of the numbers on its prime factor $$$p$$$. What is the minimum number of operations required to obtain two integers having the same number of divisors? You are given several such pairs, you need to find the answer for each of them.

The first line contains a single integer $$$t$$$ ($$$1 \le t \le 10^5$$$) — the number of pairs of integers for which you are to find the answer. Each of the next $$$t$$$ lines contain two integers $$$a_i$$$ and $$$b_i$$$ ($$$1 \le a_i, b_i \le 10^6$$$).

Output $$$t$$$ lines — the $$$i$$$-th of them should contain the answer for the pair $$$a_i$$$, $$$b_i$$$.

NoteThese are the numbers with equal number of divisors, which are optimal to obtain in the sample test case:   $$$(27, 10)$$$, 4 divisors  $$$(100, 1156)$$$, 9 divisors  $$$(220, 140)$$$, 12 divisors  $$$(17, 19)$$$, 2 divisors  $$$(12, 18)$$$, 6 divisors  $$$(50, 32)$$$, 6 divisors  $$$(224, 1925)$$$, 12 divisors Note that there can be several optimal pairs of numbers.


You have a set of $$$n$$$ weights. You know that their masses are $$$a_1$$$, $$$a_2$$$, ..., $$$a_n$$$ grams, but you don't know which of them has which mass. You can't distinguish the weights.However, your friend does know the mass of each weight. You can ask your friend to give you exactly $$$k$$$ weights with the total mass $$$m$$$ (both parameters $$$k$$$ and $$$m$$$ are chosen by you), and your friend will point to any valid subset of weights, if it is possible.You are allowed to make this query only once. Find the maximum possible number of weights you can reveal after this query.

The first line contains a single integer $$$n$$$ ($$$1 \le n \le 100$$$) — the number of weights. The second line contains $$$n$$$ integers $$$a_1, a_2, \ldots, a_n$$$ ($$$1 \le a_i \le 100$$$) — the masses of the weights.

Print the maximum number of weights you can learn the masses for after making a single query.

NoteIn the first example we can ask for a subset of two weights with total mass being equal to $$$4$$$, and the only option is to get $$$\{2, 2\}$$$.Another way to obtain the same result is to ask for a subset of two weights with the total mass of $$$5$$$ and get $$$\{1, 4\}$$$. It is easy to see that the two remaining weights have mass of $$$2$$$ grams each.In the second example we can ask for a subset of two weights with total mass being $$$8$$$, and the only answer is $$$\{4, 4\}$$$. We can prove it is not possible to learn masses for three weights in one query, but we won't put the proof here.


Alice has a lovely piece of cloth. It has the shape of a square with a side of length $$$a$$$ centimeters. Bob also wants such piece of cloth. He would prefer a square with a side of length $$$b$$$ centimeters (where $$$b < a$$$). Alice wanted to make Bob happy, so she cut the needed square out of the corner of her piece and gave it to Bob. Now she is left with an ugly L shaped cloth (see pictures below).Alice would like to know whether the area of her cloth expressed in square centimeters is prime. Could you help her to determine it?

The first line contains a number $$$t$$$ ($$$1 \leq t \leq 5$$$) — the number of test cases. Each of the next $$$t$$$ lines describes the $$$i$$$-th test case. It contains two integers $$$a$$$ and $$$b~(1 \leq b < a \leq 10^{11})$$$ — the side length of Alice's square and the side length of the square that Bob wants.

Print $$$t$$$ lines, where the $$$i$$$-th line is the answer to the $$$i$$$-th test case. Print "YES" (without quotes) if the area of the remaining piece of cloth is prime, otherwise print "NO". You can print each letter in an arbitrary case (upper or lower).

NoteThe figure below depicts the first test case. The blue part corresponds to the piece which belongs to Bob, and the red part is the piece that Alice keeps for herself. The area of the red part is $$$6^2 - 5^2 = 36 - 25 = 11$$$, which is prime, so the answer is "YES".  In the second case, the area is $$$16^2 - 13^2 = 87$$$, which is divisible by $$$3$$$.  In the third case, the area of the remaining piece is $$$61690850361^2 - 24777622630^2 = 3191830435068605713421$$$. This number is not prime because $$$3191830435068605713421 = 36913227731 \cdot 86468472991 $$$.In the last case, the area is $$$34^2 - 33^2 = 67$$$.


Little C loves number «3» very much. He loves all things about it.Now he is playing a game on a chessboard of size $$$n \times m$$$. The cell in the $$$x$$$-th row and in the $$$y$$$-th column is called $$$(x,y)$$$. Initially, The chessboard is empty. Each time, he places two chessmen on two different empty cells, the Manhattan distance between which is exactly $$$3$$$. The Manhattan distance between two cells $$$(x_i,y_i)$$$ and $$$(x_j,y_j)$$$ is defined as $$$|x_i-x_j|+|y_i-y_j|$$$.He want to place as many chessmen as possible on the chessboard. Please help him find the maximum number of chessmen he can place.

A single line contains two integers $$$n$$$ and $$$m$$$ ($$$1 \leq n,m \leq 10^9$$$) — the number of rows and the number of columns of the chessboard.

Print one integer — the maximum number of chessmen Little C can place.

NoteIn the first example, the Manhattan distance between any two cells is smaller than $$$3$$$, so the answer is $$$0$$$.In the second example, a possible solution is $$$(1,1)(3,2)$$$, $$$(1,2)(3,3)$$$, $$$(2,1)(1,3)$$$, $$$(3,1)(2,3)$$$.


You have $$$n$$$ coins, each of the same value of $$$1$$$.Distribute them into packets such that any amount $$$x$$$ ($$$1 \leq x \leq n$$$) can be formed using some (possibly one or all) number of these packets.Each packet may only be used entirely or not used at all. No packet may be used more than once in the formation of the single $$$x$$$, however it may be reused for the formation of other $$$x$$$'s.Find the minimum number of packets in such a distribution.

The only line contains a single integer $$$n$$$ ($$$1 \leq n \leq 10^9$$$) — the number of coins you have.

Output a single integer — the minimum possible number of packets, satisfying the condition above.

NoteIn the first example, three packets with $$$1$$$, $$$2$$$ and $$$3$$$ coins can be made to get any amount $$$x$$$ ($$$1\leq x\leq 6$$$).  To get $$$1$$$ use the packet with $$$1$$$ coin.  To get $$$2$$$ use the packet with $$$2$$$ coins.  To get $$$3$$$ use the packet with $$$3$$$ coins.  To get $$$4$$$ use packets with $$$1$$$ and $$$3$$$ coins.  To get $$$5$$$ use packets with $$$2$$$ and $$$3$$$ coins  To get $$$6$$$ use all packets. In the second example, two packets with $$$1$$$ and $$$1$$$ coins can be made to get any amount $$$x$$$ ($$$1\leq x\leq 2$$$).


A group of $$$n$$$ dancers rehearses a performance for the closing ceremony. The dancers are arranged in a row, they've studied their dancing moves and can't change positions. For some of them, a white dancing suit is already bought, for some of them — a black one, and for the rest the suit will be bought in the future.On the day when the suits were to be bought, the director was told that the participants of the olympiad will be happy if the colors of the suits on the scene will form a palindrome. A palindrome is a sequence that is the same when read from left to right and when read from right to left. The director liked the idea, and she wants to buy suits so that the color of the leftmost dancer's suit is the same as the color of the rightmost dancer's suit, the 2nd left is the same as 2nd right, and so on.The director knows how many burls it costs to buy a white suit, and how many burls to buy a black suit. You need to find out whether it is possible to buy suits to form a palindrome, and if it's possible, what's the minimal cost of doing so. Remember that dancers can not change positions, and due to bureaucratic reasons it is not allowed to buy new suits for the dancers who already have suits, even if it reduces the overall spending.

The first line contains three integers $$$n$$$, $$$a$$$, and $$$b$$$ ($$$1 \leq n \leq 20$$$, $$$1 \leq a, b \leq 100$$$) — the number of dancers, the cost of a white suit, and the cost of a black suit. The next line contains $$$n$$$ numbers $$$c_i$$$, $$$i$$$-th of which denotes the color of the suit of the $$$i$$$-th dancer. Number $$$0$$$ denotes the white color, $$$1$$$ — the black color, and $$$2$$$ denotes that a suit for this dancer is still to be bought.

If it is not possible to form a palindrome without swapping dancers and buying new suits for those who have one, then output -1. Otherwise, output the minimal price to get the desired visual effect.

NoteIn the first sample, the cheapest way to obtain palindromic colors is to buy a black suit for the third from left dancer and a white suit for the rightmost dancer.In the second sample, the leftmost dancer's suit already differs from the rightmost dancer's suit so there is no way to obtain the desired coloring.In the third sample, all suits are already bought and their colors form a palindrome.


Monocarp has decided to buy a new TV set and hang it on the wall in his flat. The wall has enough free space so Monocarp can buy a TV set with screen width not greater than $$$a$$$ and screen height not greater than $$$b$$$. Monocarp is also used to TV sets with a certain aspect ratio: formally, if the width of the screen is $$$w$$$, and the height of the screen is $$$h$$$, then the following condition should be met: $$$\frac{w}{h} = \frac{x}{y}$$$.There are many different TV sets in the shop. Monocarp is sure that for any pair of positive integers $$$w$$$ and $$$h$$$ there is a TV set with screen width $$$w$$$ and height $$$h$$$ in the shop.Monocarp isn't ready to choose the exact TV set he is going to buy. Firstly he wants to determine the optimal screen resolution. He has decided to try all possible variants of screen size. But he must count the number of pairs of positive integers $$$w$$$ and $$$h$$$, beforehand, such that $$$(w \le a)$$$, $$$(h \le b)$$$ and $$$(\frac{w}{h} = \frac{x}{y})$$$.In other words, Monocarp wants to determine the number of TV sets having aspect ratio $$$\frac{x}{y}$$$, screen width not exceeding $$$a$$$, and screen height not exceeding $$$b$$$. Two TV sets are considered different if they have different screen width or different screen height.

The first line contains four integers $$$a$$$, $$$b$$$, $$$x$$$, $$$y$$$ ($$$1 \le a, b, x, y \le 10^{18}$$$) — the constraints on the screen width and height, and on the aspect ratio.

Print one integer — the number of different variants to choose TV screen width and screen height so that they meet the aforementioned constraints.

NoteIn the first example, there are $$$3$$$ possible variants: $$$(5, 3)$$$, $$$(10, 6)$$$, $$$(15, 9)$$$.In the second example, there is no TV set meeting the constraints.In the third example, there is only one variant: $$$(3, 2)$$$.


Awruk is taking part in elections in his school. It is the final round. He has only one opponent — Elodreip. The are $$$n$$$ students in the school. Each student has exactly $$$k$$$ votes and is obligated to use all of them. So Awruk knows that if a person gives $$$a_i$$$ votes for Elodreip, than he will get exactly $$$k - a_i$$$ votes from this person. Of course $$$0 \le k - a_i$$$ holds.Awruk knows that if he loses his life is over. He has been speaking a lot with his friends and now he knows $$$a_1, a_2, \dots, a_n$$$ — how many votes for Elodreip each student wants to give. Now he wants to change the number $$$k$$$ to win the elections. Of course he knows that bigger $$$k$$$ means bigger chance that somebody may notice that he has changed something and then he will be disqualified.So, Awruk knows $$$a_1, a_2, \dots, a_n$$$ — how many votes each student will give to his opponent. Help him select the smallest winning number $$$k$$$. In order to win, Awruk needs to get strictly more votes than Elodreip.

The first line contains integer $$$n$$$ ($$$1 \le n \le 100$$$) — the number of students in the school. The second line contains $$$n$$$ integers $$$a_1, a_2, \ldots, a_n$$$ ($$$1 \leq a_i \leq 100$$$) — the number of votes each student gives to Elodreip.

Output the smallest integer $$$k$$$ ($$$k \ge \max a_i$$$) which gives Awruk the victory. In order to win, Awruk needs to get strictly more votes than Elodreip.

NoteIn the first example, Elodreip gets $$$1 + 1 + 1 + 5 + 1 = 9$$$ votes. The smallest possible $$$k$$$ is $$$5$$$ (it surely can't be less due to the fourth person), and it leads to $$$4 + 4 + 4 + 0 + 4 = 16$$$ votes for Awruk, which is enough to win.In the second example, Elodreip gets $$$11$$$ votes. If $$$k = 4$$$, Awruk gets $$$9$$$ votes and loses to Elodreip.


You are given an $$$n \times m$$$ grid. Each grid cell is filled with a unique integer from $$$1$$$ to $$$nm$$$ so that each integer appears exactly once.In one operation, you can choose an arbitrary cycle of the grid and move all integers along that cycle one space over. Here, a cycle is any sequence that satisfies the following conditions:  There are at least four squares.  Each square appears at most once.  Every pair of adjacent squares, and also the first and last squares, share an edge. For example, if we had the following grid:  We can choose an arbitrary cycle like this one:  To get the following grid:  In this particular case, the chosen cycle can be represented as the sequence $$$[1, 2, 3, 6, 5, 8, 7, 4]$$$, the numbers are in the direction that we want to rotate them in.Find any sequence of operations to sort the grid so that the array created by concatenating the rows from the highest to the lowest is sorted (look at the first picture above).Note you do not need to minimize number of operations or sum of cycle lengths. The only constraint is that the sum of all cycles lengths must not be greater than $$$10^5$$$. We can show that an answer always exists under the given constraints. Output any valid sequence of moves that will sort the grid.

The first line contains two integers $$$n$$$ and $$$m$$$ ($$$3 \leq n,m \leq 20$$$) — the dimensions of the grid. Each of the next $$$n$$$ lines contains $$$m$$$ integers $$$x_{i,1}, x_{i,2}, \ldots, x_{i, m}$$$ ($$$1 \leq x_{i,j} \leq nm$$$), denoting the values of the block in row $$$i$$$ and column $$$j$$$.  It is guaranteed that all $$$x_{i,j}$$$ are distinct.

First, print a single integer $$$k$$$, the number of operations ($$$k \geq 0$$$). On each of the next $$$k$$$ lines, print a cycle as follows: $$$$$$s\ y_1\ y_2\ \ldots\ y_s$$$$$$ Here, $$$s$$$ is the number of blocks to move ($$$s \geq 4$$$). Here we have block $$$y_1$$$ moving to where block $$$y_2$$$ is, block $$$y_2$$$ moving to where block $$$y_3$$$ is, and so on with block $$$y_s$$$ moving to where block $$$y_1$$$ is. The sum of $$$s$$$ over all operations must be at most $$$10^5$$$.

NoteThe first sample is the case in the statement. Here, we can use the cycle in reverse order to sort the grid.


Masha lives in a multi-storey building, where floors are numbered with positive integers. Two floors are called adjacent if their numbers differ by one. Masha decided to visit Egor. Masha lives on the floor $$$x$$$, Egor on the floor $$$y$$$ (not on the same floor with Masha).The house has a staircase and an elevator. If Masha uses the stairs, it takes $$$t_1$$$ seconds for her to walk between adjacent floors (in each direction). The elevator passes between adjacent floors (in each way) in $$$t_2$$$ seconds. The elevator moves with doors closed. The elevator spends $$$t_3$$$ seconds to open or close the doors. We can assume that time is not spent on any action except moving between adjacent floors and waiting for the doors to open or close. If Masha uses the elevator, it immediately goes directly to the desired floor.Coming out of the apartment on her floor, Masha noticed that the elevator is now on the floor $$$z$$$ and has closed doors. Now she has to choose whether to use the stairs or use the elevator. If the time that Masha needs to get to the Egor's floor by the stairs is strictly less than the time it will take her using the elevator, then she will use the stairs, otherwise she will choose the elevator.Help Mary to understand whether to use the elevator or the stairs.

The only line contains six integers $$$x$$$, $$$y$$$, $$$z$$$, $$$t_1$$$, $$$t_2$$$, $$$t_3$$$ ($$$1 \leq x, y, z, t_1, t_2, t_3 \leq 1000$$$) — the floor Masha is at, the floor Masha wants to get to, the floor the elevator is located on, the time it takes Masha to pass between two floors by stairs, the time it takes the elevator to pass between two floors and the time it takes for the elevator to close or open the doors. It is guaranteed that $$$x \ne y$$$.

If the time it will take to use the elevator is not greater than the time it will take to use the stairs, print «YES» (without quotes), otherwise print «NO> (without quotes). You can print each letter in any case (upper or lower).

NoteIn the first example:If Masha goes by the stairs, the time she spends is $$$4 \cdot 4 = 16$$$, because she has to go $$$4$$$ times between adjacent floors and each time she spends $$$4$$$ seconds. If she chooses the elevator, she will have to wait $$$2$$$ seconds while the elevator leaves the $$$4$$$-th floor and goes to the $$$5$$$-th. After that the doors will be opening for another $$$1$$$ second. Then Masha will enter the elevator, and she will have to wait for $$$1$$$ second for the doors closing. Next, the elevator will spend $$$4 \cdot 2 = 8$$$ seconds going from the $$$5$$$-th floor to the $$$1$$$-st, because the elevator has to pass $$$4$$$ times between adjacent floors and spends $$$2$$$ seconds each time. And finally, it will take another $$$1$$$ second before the doors are open and Masha can come out. Thus, all the way by elevator will take $$$2 + 1 + 1 + 8 + 1 = 13$$$ seconds, which is less than $$$16$$$ seconds, so Masha has to choose the elevator.In the second example, it is more profitable for Masha to use the stairs, because it will take $$$13$$$ seconds to use the elevator, that is more than the $$$10$$$ seconds it will takes to go by foot.In the third example, the time it takes to use the elevator is equal to the time it takes to walk up by the stairs, and is equal to $$$12$$$ seconds. That means Masha will take the elevator.


Bob and Alice are often participating in various programming competitions. Like many competitive programmers, Alice and Bob have good and bad days. They noticed, that their lucky and unlucky days are repeating with some period. For example, for Alice days $$$[l_a; r_a]$$$ are lucky, then there are some unlucky days: $$$[r_a + 1; l_a + t_a - 1]$$$, and then there are lucky days again: $$$[l_a + t_a; r_a + t_a]$$$ and so on. In other words, the day is lucky for Alice if it lies in the segment $$$[l_a + k t_a; r_a + k t_a]$$$ for some non-negative integer $$$k$$$.The Bob's lucky day have similar structure, however the parameters of his sequence are different: $$$l_b$$$, $$$r_b$$$, $$$t_b$$$. So a day is a lucky for Bob if it lies in a segment $$$[l_b + k t_b; r_b + k t_b]$$$, for some non-negative integer $$$k$$$.Alice and Bob want to participate in team competitions together and so they want to find out what is the largest possible number of consecutive days, which are lucky for both Alice and Bob.

The first line contains three integers $$$l_a$$$, $$$r_a$$$, $$$t_a$$$ ($$$0 \le l_a \le r_a \le t_a - 1, 2 \le t_a \le 10^9$$$) and describes Alice's lucky days. The second line contains three integers $$$l_b$$$, $$$r_b$$$, $$$t_b$$$ ($$$0 \le l_b \le r_b \le t_b - 1, 2 \le t_b \le 10^9$$$) and describes Bob's lucky days. It is guaranteed that both Alice and Bob have some unlucky days.

Print one integer: the maximum number of days in the row that are lucky for both Alice and Bob.

NoteThe graphs below correspond to the two sample tests and show the lucky and unlucky days of Alice and Bob as well as the possible solutions for these tests.


Arkady and his friends love playing checkers on an $$$n \times n$$$ field. The rows and the columns of the field are enumerated from $$$1$$$ to $$$n$$$.The friends have recently won a championship, so Arkady wants to please them with some candies. Remembering an old parable (but not its moral), Arkady wants to give to his friends one set of candies per each cell of the field: the set of candies for cell $$$(i, j)$$$ will have exactly $$$(i^2 + j^2)$$$ candies of unique type.There are $$$m$$$ friends who deserve the present. How many of these $$$n \times n$$$ sets of candies can be split equally into $$$m$$$ parts without cutting a candy into pieces? Note that each set has to be split independently since the types of candies in different sets are different.

The only line contains two integers $$$n$$$ and $$$m$$$ ($$$1 \le n \le 10^9$$$, $$$1 \le m \le 1000$$$) — the size of the field and the number of parts to split the sets into.

Print a single integer — the number of sets that can be split equally.

NoteIn the first example, only the set for cell $$$(3, 3)$$$ can be split equally ($$$3^2 + 3^2 = 18$$$, which is divisible by $$$m=3$$$).In the second example, the sets for the following cells can be divided equally:   $$$(1, 2)$$$ and $$$(2, 1)$$$, since $$$1^2 + 2^2 = 5$$$, which is divisible by $$$5$$$;  $$$(1, 3)$$$ and $$$(3, 1)$$$;  $$$(2, 4)$$$ and $$$(4, 2)$$$;  $$$(2, 6)$$$ and $$$(6, 2)$$$;  $$$(3, 4)$$$ and $$$(4, 3)$$$;  $$$(3, 6)$$$ and $$$(6, 3)$$$;  $$$(5, 5)$$$. In the third example, sets in all cells can be divided equally, since $$$m = 1$$$.


Having problems with tram routes in the morning, Arkady decided to return home by metro. Fortunately for Arkady, there is only one metro line in the city.Unfortunately for Arkady, the line is circular. It means that the stations are enumerated from $$$1$$$ to $$$n$$$ and there is a tunnel between any pair of consecutive stations as well as between the station $$$1$$$ and the station $$$n$$$. Trains that go in clockwise direction visit the stations in order $$$1 \to 2 \to 3 \to \ldots \to n \to 1$$$ while the trains that go in the counter-clockwise direction visit the stations in the reverse order.The stations that have numbers from $$$1$$$ to $$$m$$$ have interior mostly in red colors while the stations that have numbers from $$$m + 1$$$ to $$$n$$$ have blue interior. Arkady entered the metro at station $$$s$$$ and decided to use the following algorithm to choose his way home.  Initially he has a positive integer $$$t$$$ in his mind.  If the current station has red interior, he takes a clockwise-directed train, otherwise he takes a counter-clockwise-directed train.  He rides exactly $$$t$$$ stations on the train and leaves the train.  He decreases $$$t$$$ by one. If $$$t$$$ is still positive, he returns to step $$$2$$$. Otherwise he exits the metro. You have already realized that this algorithm most probably won't bring Arkady home. Find the station he will exit the metro at so that you can continue helping him.

The first line contains two integers $$$n$$$ and $$$m$$$ ($$$3 \le n \le 10^5$$$, $$$1 \le m < n$$$) — the total number of stations and the number of stations that have red interior. The second line contains two integers $$$s$$$ and $$$t$$$ ($$$1 \le s \le n$$$, $$$1 \le t \le 10^{12}$$$) — the starting station and the initial value of $$$t$$$.

Output the only integer — the station where Arkady will exit the metro.

NoteConsider the first example. There are $$$10$$$ stations and the first $$$4$$$ of them are red. Arkady starts at station $$$3$$$ with value $$$t = 1$$$, so just rides $$$1$$$ station in clockwise direction and ends up on the station $$$4$$$.In the second example the metro is same as in the first example, but Arkady starts at station $$$3$$$ with value $$$t = 5$$$.   It is a red station so he rides $$$5$$$ stations in clockwise direction and leaves the train at station $$$8$$$.  It is a blue station, so he rides $$$4$$$ stations in counter-clockwise direction and leaves at station $$$4$$$.  It is a red station, so he rides $$$3$$$ stations in clockwise direction and leaves at station $$$7$$$.  It is a blue station, so he rides $$$2$$$ stations in counter-clockwise direction and leaves at station $$$5$$$.  It is a blue station, so he rides $$$1$$$ station in counter-clockwise direction and leaves at station $$$4$$$.  Now $$$t = 0$$$, so Arkady exits metro at the station $$$4$$$.


There are $$$n$$$ candy boxes in front of Tania. The boxes are arranged in a row from left to right, numbered from $$$1$$$ to $$$n$$$. The $$$i$$$-th box contains $$$r_i$$$ candies, candies have the color $$$c_i$$$ (the color can take one of three values — red, green, or blue). All candies inside a single box have the same color (and it is equal to $$$c_i$$$).Initially, Tanya is next to the box number $$$s$$$. Tanya can move to the neighbor box (that is, with a number that differs by one) or eat candies in the current box. Tanya eats candies instantly, but the movement takes one second.If Tanya eats candies from the box, then the box itself remains in place, but there is no more candies in it. In other words, Tanya always eats all the candies from the box and candies in the boxes are not refilled.It is known that Tanya cannot eat candies of the same color one after another (that is, the colors of candies in two consecutive boxes from which she eats candies are always different). In addition, Tanya's appetite is constantly growing, so in each next box from which she eats candies, there should be strictly more candies than in the previous one.Note that for the first box from which Tanya will eat candies, there are no restrictions on the color and number of candies.Tanya wants to eat at least $$$k$$$ candies. What is the minimum number of seconds she will need? Remember that she eats candies instantly, and time is spent only on movements.

The first line contains three integers $$$n$$$, $$$s$$$ and $$$k$$$ ($$$1 \le n \le 50$$$, $$$1 \le s \le n$$$, $$$1 \le k \le 2000$$$) — number of the boxes, initial position of Tanya and lower bound on number of candies to eat. The following line contains $$$n$$$ integers $$$r_i$$$ ($$$1 \le r_i \le 50$$$) — numbers of candies in the boxes. The third line contains sequence of $$$n$$$ letters 'R', 'G' and 'B', meaning the colors of candies in the correspondent boxes ('R' for red, 'G' for green, 'B' for blue). Recall that each box contains candies of only one color. The third line contains no spaces.

Print minimal number of seconds to eat at least $$$k$$$ candies. If solution doesn't exist, print "-1".

NoteThe sequence of actions of Tanya for the first example:  move from the box $$$3$$$ to the box $$$2$$$;  eat candies from the box $$$2$$$;  move from the box $$$2$$$ to the box $$$3$$$;  eat candy from the box $$$3$$$;  move from the box $$$3$$$ to the box $$$4$$$;  move from the box $$$4$$$ to the box $$$5$$$;  eat candies from the box $$$5$$$. Since Tanya eats candy instantly, the required time is four seconds.


Let's call a string a phone number if it has length 11 and fits the pattern "8xxxxxxxxxx", where each "x" is replaced by a digit.For example, "80123456789" and "80000000000" are phone numbers, while "8012345678" and "79000000000" are not.You have $$$n$$$ cards with digits, and you want to use them to make as many phone numbers as possible. Each card must be used in at most one phone number, and you don't have to use all cards. The phone numbers do not necessarily have to be distinct.

The first line contains an integer $$$n$$$ — the number of cards with digits that you have ($$$1 \leq n \leq 100$$$). The second line contains a string of $$$n$$$ digits (characters "0", "1", ..., "9") $$$s_1, s_2, \ldots, s_n$$$. The string will not contain any other characters, such as leading or trailing spaces.

If at least one phone number can be made from these cards, output the maximum number of phone numbers that can be made. Otherwise, output 0.

NoteIn the first example, one phone number, "8000000000", can be made from these cards.In the second example, you can make two phone numbers from the cards, for example, "80123456789" and "80123456789".In the third example you can't make any phone number from the given cards.


You are given a positive integer $$$n$$$.Let $$$S(x)$$$ be sum of digits in base 10 representation of $$$x$$$, for example, $$$S(123) = 1 + 2 + 3 = 6$$$, $$$S(0) = 0$$$.Your task is to find two integers $$$a, b$$$, such that $$$0 \leq a, b \leq n$$$, $$$a + b = n$$$ and $$$S(a) + S(b)$$$ is the largest possible among all such pairs.

The only line of input contains an integer $$$n$$$ $$$(1 \leq n \leq 10^{12})$$$.

Print largest $$$S(a) + S(b)$$$ among all pairs of integers $$$a, b$$$, such that $$$0 \leq a, b \leq n$$$ and $$$a + b = n$$$.

NoteIn the first example, you can choose, for example, $$$a = 17$$$ and $$$b = 18$$$, so that $$$S(17) + S(18) = 1 + 7 + 1 + 8 = 17$$$. It can be shown that it is impossible to get a larger answer.In the second test example, you can choose, for example, $$$a = 5000000001$$$ and $$$b = 4999999999$$$, with $$$S(5000000001) + S(4999999999) = 91$$$. It can be shown that it is impossible to get a larger answer.


You have unlimited number of coins with values $$$1, 2, \ldots, n$$$. You want to select some set of coins having the total value of $$$S$$$. It is allowed to have multiple coins with the same value in the set. What is the minimum number of coins required to get sum $$$S$$$?

The only line of the input contains two integers $$$n$$$ and $$$S$$$ ($$$1 \le n \le 100\,000$$$, $$$1 \le S \le 10^9$$$)

Print exactly one integer — the minimum number of coins required to obtain sum $$$S$$$.

NoteIn the first example, some of the possible ways to get sum $$$11$$$ with $$$3$$$ coins are:   $$$(3, 4, 4)$$$ $$$(2, 4, 5)$$$ $$$(1, 5, 5)$$$ $$$(3, 3, 5)$$$ It is impossible to get sum $$$11$$$ with less than $$$3$$$ coins.In the second example, some of the possible ways to get sum $$$16$$$ with $$$3$$$ coins are:   $$$(5, 5, 6)$$$ $$$(4, 6, 6)$$$ It is impossible to get sum $$$16$$$ with less than $$$3$$$ coins.


JATC's math teacher always gives the class some interesting math problems so that they don't get bored. Today the problem is as follows. Given an integer $$$n$$$, you can perform the following operations zero or more times:  mul $$$x$$$: multiplies $$$n$$$ by $$$x$$$ (where $$$x$$$ is an arbitrary positive integer).  sqrt: replaces $$$n$$$ with $$$\sqrt{n}$$$ (to apply this operation, $$$\sqrt{n}$$$ must be an integer). You can perform these operations as many times as you like. What is the minimum value of $$$n$$$, that can be achieved and what is the minimum number of operations, to achieve that minimum value?Apparently, no one in the class knows the answer to this problem, maybe you can help them?

The only line of the input contains a single integer $$$n$$$ ($$$1 \le n \le 10^6$$$) — the initial number.

Print two integers: the minimum integer $$$n$$$ that can be achieved using the described operations and the minimum number of operations required.

NoteIn the first example, you can apply the operation mul $$$5$$$ to get $$$100$$$ and then sqrt to get $$$10$$$.In the second example, you can first apply sqrt to get $$$72$$$, then mul $$$18$$$ to get $$$1296$$$ and finally two more sqrt and you get $$$6$$$.Note, that even if the initial value of $$$n$$$ is less or equal $$$10^6$$$, it can still become greater than $$$10^6$$$ after applying one or more operations.


JATC and his friend Giraffe are currently in their room, solving some problems. Giraffe has written on the board an array $$$a_1$$$, $$$a_2$$$, ..., $$$a_n$$$ of integers, such that $$$1 \le a_1 < a_2 < \ldots < a_n \le 10^3$$$, and then went to the bathroom.JATC decided to prank his friend by erasing some consecutive elements in the array. Since he doesn't want for the prank to go too far, he will only erase in a way, such that Giraffe can still restore the array using the information from the remaining elements. Because Giraffe has created the array, he's also aware that it's an increasing array and all the elements are integers in the range $$$[1, 10^3]$$$.JATC wonders what is the greatest number of elements he can erase?

The first line of the input contains a single integer $$$n$$$ ($$$1 \le n \le 100$$$) — the number of elements in the array. The second line of the input contains $$$n$$$ integers $$$a_i$$$ ($$$1 \le a_1<a_2<\dots<a_n \le 10^3$$$) — the array written by Giraffe.

Print a single integer — the maximum number of consecutive elements in the array that JATC can erase. If it is impossible to erase even a single element, print $$$0$$$.

NoteIn the first example, JATC can erase the third and fourth elements, leaving the array $$$[1, 3, \_, \_, 6, 9]$$$. As you can see, there is only one way to fill in the blanks.In the second example, JATC can erase the second and the third elements. The array will become $$$[998, \_, \_]$$$. Because all the elements are less than or equal to $$$1000$$$, the array is still can be restored. Note, that he can't erase the first $$$2$$$ elements.In the third example, JATC can erase the first $$$4$$$ elements. Since all the elements are greater than or equal to $$$1$$$, Giraffe can still restore the array. Note, that he can't erase the last $$$4$$$ elements.


At the children's festival, children were dancing in a circle. When music stopped playing, the children were still standing in a circle. Then Lena remembered, that her parents gave her a candy box with exactly $$$k$$$ candies "Wilky May". Lena is not a greedy person, so she decided to present all her candies to her friends in the circle. Lena knows, that some of her friends have a sweet tooth and others do not. Sweet tooth takes out of the box two candies, if the box has at least two candies, and otherwise takes one. The rest of Lena's friends always take exactly one candy from the box.Before starting to give candies, Lena step out of the circle, after that there were exactly $$$n$$$ people remaining there. Lena numbered her friends in a clockwise order with positive integers starting with $$$1$$$ in such a way that index $$$1$$$ was assigned to her best friend Roma.Initially, Lena gave the box to the friend with number $$$l$$$, after that each friend (starting from friend number $$$l$$$) took candies from the box and passed the box to the next friend in clockwise order. The process ended with the friend number $$$r$$$ taking the last candy (or two, who knows) and the empty box. Please note that it is possible that some of Lena's friends took candy from the box several times, that is, the box could have gone several full circles before becoming empty.Lena does not know which of her friends have a sweet tooth, but she is interested in the maximum possible number of friends that can have a sweet tooth. If the situation could not happen, and Lena have been proved wrong in her observations, please tell her about this.

The only line contains four integers $$$n$$$, $$$l$$$, $$$r$$$ and $$$k$$$ ($$$1 \le n, k \le 10^{11}$$$, $$$1 \le l, r \le n$$$) — the number of children in the circle, the number of friend, who was given a box with candies, the number of friend, who has taken last candy and the initial number of candies in the box respectively.

Print exactly one integer — the maximum possible number of sweet tooth among the friends of Lena or "-1" (quotes for clarity), if Lena is wrong.

NoteIn the first example, any two friends can be sweet tooths, this way each person will receive the box with candies twice and the last person to take sweets will be the fourth friend.In the second example, sweet tooths can be any three friends, except for the friend on the third position.In the third example, only one friend will take candy, but he can still be a sweet tooth, but just not being able to take two candies. All other friends in the circle can be sweet tooths as well, they just will not be able to take a candy even once.In the fourth example, Lena is wrong and this situation couldn't happen.


Vasya has got an undirected graph consisting of $$$n$$$ vertices and $$$m$$$ edges. This graph doesn't contain any self-loops or multiple edges. Self-loop is an edge connecting a vertex to itself. Multiple edges are a pair of edges such that they connect the same pair of vertices. Since the graph is undirected, the pair of edges $$$(1, 2)$$$ and $$$(2, 1)$$$ is considered to be multiple edges. Isolated vertex of the graph is a vertex such that there is no edge connecting this vertex to any other vertex.Vasya wants to know the minimum and maximum possible number of isolated vertices in an undirected graph consisting of $$$n$$$ vertices and $$$m$$$ edges.

The only line contains two integers $$$n$$$ and $$$m~(1 \le n \le 10^5, 0 \le m \le \frac{n (n - 1)}{2})$$$. It is guaranteed that there exists a graph without any self-loops or multiple edges with such number of vertices and edges.

In the only line print two numbers $$$min$$$ and $$$max$$$ — the minimum and maximum number of isolated vertices, respectively.

NoteIn the first example it is possible to construct a graph with $$$0$$$ isolated vertices: for example, it should contain edges $$$(1, 2)$$$ and $$$(3, 4)$$$. To get one isolated vertex, we may construct a graph with edges $$$(1, 2)$$$ and $$$(1, 3)$$$. In the second example the graph will always contain exactly one isolated vertex.


You stumbled upon a new kind of chess puzzles. The chessboard you are given is not necesserily $$$8 \times 8$$$, but it still is $$$N \times N$$$. Each square has some number written on it, all the numbers are from $$$1$$$ to $$$N^2$$$ and all the numbers are pairwise distinct. The $$$j$$$-th square in the $$$i$$$-th row has a number $$$A_{ij}$$$ written on it.In your chess set you have only three pieces: a knight, a bishop and a rook. At first, you put one of them on the square with the number $$$1$$$ (you can choose which one). Then you want to reach square $$$2$$$ (possibly passing through some other squares in process), then square $$$3$$$ and so on until you reach square $$$N^2$$$. In one step you are allowed to either make a valid move with the current piece or replace it with some other piece. Each square can be visited arbitrary number of times.A knight can move to a square that is two squares away horizontally and one square vertically, or two squares vertically and one square horizontally. A bishop moves diagonally. A rook moves horizontally or vertically. The move should be performed to a different square from the one a piece is currently standing on.You want to minimize the number of steps of the whole traversal. Among all the paths to have the same number of steps you want to choose the one with the lowest number of piece replacements.What is the path you should take to satisfy all conditions?

The first line contains a single integer $$$N$$$ ($$$3 \le N \le 10$$$) — the size of the chessboard. Each of the next $$$N$$$ lines contains $$$N$$$ integers $$$A_{i1}, A_{i2}, \dots, A_{iN}$$$ ($$$1 \le A_{ij} \le N^2$$$) — the numbers written on the squares of the $$$i$$$-th row of the board. It is guaranteed that all $$$A_{ij}$$$ are pairwise distinct.

The only line should contain two integers — the number of steps in the best answer and the number of replacement moves in it.

NoteHere are the steps for the first example (the starting piece is a knight):  Move to $$$(3, 2)$$$  Move to $$$(1, 3)$$$  Move to $$$(3, 2)$$$  Replace the knight with a rook  Move to $$$(3, 1)$$$  Move to $$$(3, 3)$$$  Move to $$$(3, 2)$$$  Move to $$$(2, 2)$$$  Move to $$$(2, 3)$$$  Move to $$$(2, 1)$$$  Move to $$$(1, 1)$$$  Move to $$$(1, 2)$$$