Ksenia has her winter exams. Today she is learning combinatorics. Here's one of the problems she needs to learn to solve.How many distinct trees are there consisting of n vertices, each with the following properties:  the tree is marked, that is, the vertices of the tree are numbered from 1 to n;  each vertex of the tree is connected with at most three other vertices, and at the same moment the vertex with number 1 is connected with at most two other vertices;  the size of the tree's maximum matching equals k. Two trees are considered distinct if there are such two vertices u and v, that in one tree they are connected by an edge and in the other tree they are not.Help Ksenia solve the problem for the given n and k. As the answer to the problem can be very huge you should output it modulo 1000000007 (109 + 7).

The first line contains two integers n, k (1 ≤ n, k ≤ 50).

Print a single integer — the answer to the problem modulo 1000000007 (109 + 7).

NoteIf you aren't familiar with matchings, please, read the following link: http://en.wikipedia.org/wiki/Matching_(graph_theory).


Last week, Hamed learned about a new type of equations in his math class called Modular Equations. Lets define i modulo j as the remainder of division of i by j and denote it by . A Modular Equation, as Hamed's teacher described, is an equation of the form  in which a and b are two non-negative integers and x is a variable. We call a positive integer x for which  a solution of our equation.Hamed didn't pay much attention to the class since he was watching a movie. He only managed to understand the definitions of these equations.Now he wants to write his math exercises but since he has no idea how to do that, he asked you for help. He has told you all he knows about Modular Equations and asked you to write a program which given two numbers a and b determines how many answers the Modular Equation  has.

In the only line of the input two space-separated integers a and b (0 ≤ a, b ≤ 109) are given.

If there is an infinite number of answers to our equation, print "infinity" (without the quotes). Otherwise print the number of solutions of the Modular Equation .

NoteIn the first sample the answers of the Modular Equation are 8 and 16 since


There are three doors in front of you, numbered from $$$1$$$ to $$$3$$$ from left to right. Each door has a lock on it, which can only be opened with a key with the same number on it as the number on the door.There are three keys — one for each door. Two of them are hidden behind the doors, so that there is no more than one key behind each door. So two doors have one key behind them, one door doesn't have a key behind it. To obtain a key hidden behind a door, you should first unlock that door. The remaining key is in your hands.Can you open all the doors?

The first line contains a single integer $$$t$$$ ($$$1 \le t \le 18$$$) — the number of testcases. The first line of each testcase contains a single integer $$$x$$$ ($$$1 \le x \le 3$$$) — the number on the key in your hands. The second line contains three integers $$$a, b$$$ and $$$c$$$ ($$$0 \le a, b, c \le 3$$$) — the number on the key behind each of the doors. If there is no key behind the door, the number is equal to $$$0$$$. Values $$$1, 2$$$ and $$$3$$$ appear exactly once among $$$x, a, b$$$ and $$$c$$$.

For each testcase, print "YES" if you can open all the doors. Otherwise, print "NO".


Manao is trying to open a rather challenging lock. The lock has n buttons on it and to open it, you should press the buttons in a certain order to open the lock. When you push some button, it either stays pressed into the lock (that means that you've guessed correctly and pushed the button that goes next in the sequence), or all pressed buttons return to the initial position. When all buttons are pressed into the lock at once, the lock opens.Consider an example with three buttons. Let's say that the opening sequence is: {2, 3, 1}. If you first press buttons 1 or 3, the buttons unpress immediately. If you first press button 2, it stays pressed. If you press 1 after 2, all buttons unpress. If you press 3 after 2, buttons 3 and 2 stay pressed. As soon as you've got two pressed buttons, you only need to press button 1 to open the lock.Manao doesn't know the opening sequence. But he is really smart and he is going to act in the optimal way. Calculate the number of times he's got to push a button in order to open the lock in the worst-case scenario.

A single line contains integer n (1 ≤ n ≤ 2000) — the number of buttons the lock has.

In a single line print the number of times Manao has to push a button in the worst-case scenario.

NoteConsider the first test sample. Manao can fail his first push and push the wrong button. In this case he will already be able to guess the right one with his second push. And his third push will push the second right button. Thus, in the worst-case scenario he will only need 3 pushes.


Kirill plays a new computer game. He came to the potion store where he can buy any potion. Each potion is characterized by two integers — amount of experience and cost. The efficiency of a potion is the ratio of the amount of experience to the cost. Efficiency may be a non-integer number.For each two integer numbers a and b such that l ≤ a ≤ r and x ≤ b ≤ y there is a potion with experience a and cost b in the store (that is, there are (r - l + 1)·(y - x + 1) potions).Kirill wants to buy a potion which has efficiency k. Will he be able to do this?

First string contains five integer numbers l, r, x, y, k (1 ≤ l ≤ r ≤ 107, 1 ≤ x ≤ y ≤ 107, 1 ≤ k ≤ 107).

Print "YES" without quotes if a potion with efficiency exactly k can be bought in the store and "NO" without quotes otherwise. You can output each of the letters in any register.


There is a card game called "Durak", which means "Fool" in Russian. The game is quite popular in the countries that used to form USSR. The problem does not state all the game's rules explicitly — you can find them later yourselves if you want.To play durak you need a pack of 36 cards. Each card has a suit ("S", "H", "D" and "C") and a rank (in the increasing order "6", "7", "8", "9", "T", "J", "Q", "K" and "A"). At the beginning of the game one suit is arbitrarily chosen as trump. The players move like that: one player puts one or several of his cards on the table and the other one should beat each of them with his cards.A card beats another one if both cards have similar suits and the first card has a higher rank then the second one. Besides, a trump card can beat any non-trump card whatever the cards’ ranks are. In all other cases you can not beat the second card with the first one.You are given the trump suit and two different cards. Determine whether the first one beats the second one or not.

The first line contains the tramp suit. It is "S", "H", "D" or "C". The second line contains the description of the two different cards. Each card is described by one word consisting of two symbols. The first symbol stands for the rank ("6", "7", "8", "9", "T", "J", "Q", "K" and "A"), and the second one stands for the suit ("S", "H", "D" and "C").

Print "YES" (without the quotes) if the first cards beats the second one. Otherwise, print "NO" (also without the quotes).


The king's birthday dinner was attended by $$$k$$$ guests. The dinner was quite a success: every person has eaten several dishes (though the number of dishes was the same for every person) and every dish was served alongside with a new set of kitchen utensils.All types of utensils in the kingdom are numbered from $$$1$$$ to $$$100$$$. It is known that every set of utensils is the same and consist of different types of utensils, although every particular type may appear in the set at most once. For example, a valid set of utensils can be composed of one fork, one spoon and one knife.After the dinner was over and the guests were dismissed, the king wondered what minimum possible number of utensils could be stolen. Unfortunately, the king has forgotten how many dishes have been served for every guest but he knows the list of all the utensils left after the dinner. Your task is to find the minimum possible number of stolen utensils.

The first line contains two integer numbers $$$n$$$ and $$$k$$$ ($$$1 \le n \le 100, 1 \le k \le 100$$$)  — the number of kitchen utensils remaining after the dinner and the number of guests correspondingly. The next line contains $$$n$$$ integers $$$a_1, a_2, \ldots, a_n$$$ ($$$1 \le a_i \le 100$$$)  — the types of the utensils remaining. Equal values stand for identical utensils while different values stand for different utensils.

Output a single value — the minimum number of utensils that could be stolen by the guests.

NoteIn the first example it is clear that at least one utensil of type $$$3$$$ has been stolen, since there are two guests and only one such utensil. But it is also possible that every person received only one dish and there were only six utensils in total, when every person got a set $$$(1, 2, 3)$$$ of utensils. Therefore, the answer is $$$1$$$.One can show that in the second example at least $$$2$$$ dishes should have been served for every guest, so the number of utensils should be at least $$$24$$$: every set contains $$$4$$$ utensils and every one of the $$$3$$$ guests gets two such sets. Therefore, at least $$$14$$$ objects have been stolen. Please note that utensils of some types (for example, of types $$$2$$$ and $$$4$$$ in this example) may be not present in the set served for dishes.


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.


Consider a linear function f(x) = Ax + B. Let's define g(0)(x) = x and g(n)(x) = f(g(n - 1)(x)) for n > 0. For the given integer values A, B, n and x find the value of g(n)(x) modulo 109 + 7.

The only line contains four integers A, B, n and x (1 ≤ A, B, x ≤ 109, 1 ≤ n ≤ 1018) — the parameters from the problem statement. Note that the given value n can be too large, so you should use 64-bit integer type to store it. In C++ you can use the long long integer type and in Java you can use long integer type.

Print the only integer s — the value g(n)(x) modulo 109 + 7.


Vasya plays The Elder Trolls IV: Oblivon. Oh, those creators of computer games! What they do not come up with! Absolutely unique monsters have been added to the The Elder Trolls IV: Oblivon. One of these monsters is Unkillable Slug. Why it is "Unkillable"? Firstly, because it can be killed with cutting weapon only, so lovers of two-handed amber hammers should find suitable knife themselves. Secondly, it is necessary to make so many cutting strokes to Unkillable Slug. Extremely many. Too many! Vasya has already promoted his character to 80-th level and in order to gain level 81 he was asked to kill Unkillable Slug. The monster has a very interesting shape. It looks like a rectangular parallelepiped with size x × y × z, consisting of undestructable cells 1 × 1 × 1. At one stroke Vasya can cut the Slug along an imaginary grid, i.e. cut with a plane parallel to one of the parallelepiped side. Monster dies when amount of parts it is divided reaches some critical value.All parts of monster do not fall after each cut, they remains exactly on its places. I. e. Vasya can cut several parts with one cut.Vasya wants to know what the maximum number of pieces he can cut the Unkillable Slug into striking him at most k times.Vasya's character uses absolutely thin sword with infinite length.

The first line of input contains four integer numbers x, y, z, k (1 ≤ x, y, z ≤ 106, 0 ≤ k ≤ 109).

Output the only number — the answer for the problem. Please, do not use %lld specificator to read or write 64-bit integers in C++. It is preffered to use cout (also you may use %I64d).

NoteIn the first sample Vasya make 3 pairwise perpendicular cuts. He cuts monster on two parts with the first cut, then he divides each part on two with the second cut, and finally he divides each of the 4 parts on two.


Fibonacci strings are defined as follows:   f1 = «a»  f2 = «b»  fn = fn - 1 fn - 2, n > 2 Thus, the first five Fibonacci strings are: "a", "b", "ba", "bab", "babba".You are given a Fibonacci string and m strings si. For each string si, find the number of times it occurs in the given Fibonacci string as a substring.

The first line contains two space-separated integers k and m — the number of a Fibonacci string and the number of queries, correspondingly. Next m lines contain strings si that correspond to the queries. It is guaranteed that strings si aren't empty and consist only of characters "a" and "b". The input limitations for getting 30 points are:    1 ≤ k ≤ 3000  1 ≤ m ≤ 3000  The total length of strings si doesn't exceed 3000  The input limitations for getting 100 points are:    1 ≤ k ≤ 1018  1 ≤ m ≤ 104  The total length of strings si doesn't exceed 105  Please do not use the %lld specifier to read or write 64-bit integers in С++. It is preferred to use cin, cout streams or the %I64d specifier.

For each string si print the number of times it occurs in the given Fibonacci string as a substring. Since the numbers can be large enough, print them modulo 1000000007 (109 + 7). Print the answers for the strings in the order in which they are given in the input.


The Duck songFor simplicity, we'll assume that there are only three types of grapes: green grapes, purple grapes and black grapes.Andrew, Dmitry and Michal are all grapes' lovers, however their preferences of grapes are different. To make all of them happy, the following should happen: Andrew, Dmitry and Michal should eat at least $$$x$$$, $$$y$$$ and $$$z$$$ grapes, respectively. Andrew has an extreme affinity for green grapes, thus he will eat green grapes and green grapes only. On the other hand, Dmitry is not a fan of black grapes — any types of grapes except black would do for him. In other words, Dmitry can eat green and purple grapes. Michal has a common taste — he enjoys grapes in general and will be pleased with any types of grapes, as long as the quantity is sufficient.Knowing that his friends are so fond of grapes, Aki decided to host a grape party with them. He has prepared a box with $$$a$$$ green grapes, $$$b$$$ purple grapes and $$$c$$$ black grapes.However, Aki isn't sure if the box he prepared contains enough grapes to make everyone happy. Can you please find out whether it's possible to distribute grapes so that everyone is happy or Aki has to buy some more grapes?It is not required to distribute all the grapes, so it's possible that some of them will remain unused.

The first line contains three integers $$$x$$$, $$$y$$$ and $$$z$$$ ($$$1 \le x, y, z \le 10^5$$$) — the number of grapes Andrew, Dmitry and Michal want to eat. The second line contains three integers $$$a$$$, $$$b$$$, $$$c$$$ ($$$1 \le a, b, c \le 10^5$$$) — the number of green, purple and black grapes in the box.

If there is a grape distribution that allows everyone to be happy, print "YES", otherwise print "NO".

NoteIn the first example, there is only one possible distribution:Andrew should take $$$1$$$ green grape, Dmitry should take $$$3$$$ remaining green grapes and $$$3$$$ purple grapes, and Michal will take $$$2$$$ out of $$$3$$$ available black grapes.In the second test, there is no possible distribution, since Andrew is not be able to eat enough green grapes. :(


Linda likes to change her hair color from time to time, and would be pleased if her boyfriend Archie would notice the difference between the previous and the new color. Archie always comments on Linda's hair color if and only if he notices a difference — so Linda always knows whether Archie has spotted the difference or not.There is a new hair dye series in the market where all available colors are numbered by integers from $$$1$$$ to $$$N$$$ such that a smaller difference of the numerical values also means less visual difference.Linda assumes that for these series there should be some critical color difference $$$C$$$ ($$$1 \le C \le N$$$) for which Archie will notice color difference between the current color $$$\mathrm{color}_{\mathrm{new}}$$$ and the previous color $$$\mathrm{color}_{\mathrm{prev}}$$$ if $$$\left|\mathrm{color}_{\mathrm{new}} - \mathrm{color}_{\mathrm{prev}}\right| \ge C$$$ and will not if $$$\left|\mathrm{color}_{\mathrm{new}} - \mathrm{color}_{\mathrm{prev}}\right| < C$$$.Now she has bought $$$N$$$ sets of hair dye from the new series — one for each of the colors from $$$1$$$ to $$$N$$$, and is ready to set up an experiment. Linda will change her hair color on a regular basis and will observe Archie's reaction — whether he will notice the color change or not. Since for the proper dye each set should be used completely, each hair color can be obtained no more than once.Before the experiment, Linda was using a dye from a different series which is not compatible with the new one, so for the clearness of the experiment Archie's reaction to the first used color is meaningless.Her aim is to find the precise value of $$$C$$$ in a limited number of dyes. Write a program which finds the value of $$$C$$$ by experimenting with the given $$$N$$$ colors and observing Archie's reactions to color changes.

NoteComments to the example input line by line:   $$$N = 7$$$.  Answer to the first query is meaningless (can also be $$$0$$$).  $$$C \leq 5$$$.  $$$3 < C \leq 5$$$. It would be wise to check difference $$$4$$$. However, this can not be done in the next query since $$$4 + 4 = 8$$$ and $$$4 - 4 = 0$$$ both are outside the allowed interval $$$1 \le P \le 7$$$.  $$$3 < C \leq 5$$$.  $$$3 < C \leq 4$$$. Therefore, $$$C = 4$$$.


To quickly hire highly skilled specialists one of the new IT City companies made an unprecedented move. Every employee was granted a car, and an employee can choose one of four different car makes.The parking lot before the office consists of one line of (2n - 2) parking spaces. Unfortunately the total number of cars is greater than the parking lot capacity. Furthermore even amount of cars of each make is greater than the amount of parking spaces! That's why there are no free spaces on the parking lot ever.Looking on the straight line of cars the company CEO thought that parking lot would be more beautiful if it contained exactly n successive cars of the same make. Help the CEO determine the number of ways to fill the parking lot this way.

The only line of the input contains one integer n (3 ≤ n ≤ 30) — the amount of successive cars of the same make.

Output one integer — the number of ways to fill the parking lot by cars of four makes using the described way.

NoteLet's denote car makes in the following way: A — Aston Martin, B —   Bentley, M — Mercedes-Maybach, Z — Zaporozhets. For n = 3 there are the following appropriate ways to fill the parking lot: AAAB AAAM AAAZ ABBB AMMM AZZZ BBBA BBBM BBBZ BAAA BMMM BZZZ MMMA MMMB MMMZ MAAA MBBB MZZZ ZZZA ZZZB ZZZM ZAAA ZBBB ZMMMOriginally it was planned to grant sport cars of Ferrari, Lamborghini, Maserati and Bugatti makes but this idea was renounced because it is impossible to drive these cars having small road clearance on the worn-down roads of IT City.


Everyone loves a freebie. Especially students.It is well-known that if in the night before exam a student opens window, opens the student's record-book and shouts loudly three times "Fly, freebie, fly!" — then flown freebie helps him to pass the upcoming exam.In the night before the exam on mathematical analysis n students living in dormitory shouted treasured words. The i-th student made a sacrament at the time ti, where ti is the number of seconds elapsed since the beginning of the night.It is known that the freebie is a capricious and willful lady. That night the freebie was near dormitory only for T seconds. Therefore, if for two students their sacrament times differ for more than T, then the freebie didn't visit at least one of them.Since all students are optimists, they really want to know what is the maximal number of students visited by the freebie can be.

The first line of the input contains integer n (1 ≤ n ≤ 100), where n — the number of students shouted "Fly, freebie, fly!" The second line contains n positive integers ti (1 ≤ ti ≤ 1000). The last line contains integer T (1 ≤ T ≤ 1000) — the time interval during which the freebie was near the dormitory.

Print a single integer — the largest number of people who will pass exam tomorrow because of the freebie visit.


Today is the final contest of INOI (Iranian National Olympiad in Informatics). The contest room is a row with $$$n$$$ computers. All computers are numbered with integers from $$$1$$$ to $$$n$$$ from left to right. There are $$$m$$$ participants, numbered with integers from $$$1$$$ to $$$m$$$.We have an array $$$a$$$ of length $$$m$$$ where $$$a_{i}$$$ ($$$1 \leq a_i \leq n$$$) is the computer behind which the $$$i$$$-th participant wants to sit.Also, we have another array $$$b$$$ of length $$$m$$$ consisting of characters 'L' and 'R'. $$$b_i$$$ is the side from which the $$$i$$$-th participant enters the room. 'L' means the participant enters from the left of computer $$$1$$$ and goes from left to right, and 'R' means the participant enters from the right of computer $$$n$$$ and goes from right to left.The participants in the order from $$$1$$$ to $$$m$$$ enter the room one by one. The $$$i$$$-th of them enters the contest room in the direction $$$b_i$$$ and goes to sit behind the $$$a_i$$$-th computer. If it is occupied he keeps walking in his direction until he reaches the first unoccupied computer. After that, he sits behind it. If he doesn't find any computer he gets upset and gives up on the contest.The madness of the $$$i$$$-th participant is the distance between his assigned computer ($$$a_i$$$) and the computer he ends up sitting behind. The distance between computers $$$i$$$ and $$$j$$$ is equal to $$$|i - j|$$$.The values in the array $$$a$$$ can be equal. There exist $$$n^m \cdot 2^m$$$ possible pairs of arrays $$$(a, b)$$$.Consider all pairs of arrays $$$(a, b)$$$ such that no person becomes upset. For each of them let's calculate the sum of participants madnesses. Find the sum of all these values.You will be given some prime modulo $$$p$$$. Find this sum by modulo $$$p$$$.

The only line contains three integers $$$n$$$, $$$m$$$, $$$p$$$ ($$$1 \leq m \leq n \leq 500, 10^8 \leq p \leq 10 ^ 9 + 9$$$). It is guaranteed, that the number $$$p$$$ is prime.

Print only one integer — the required sum by modulo $$$p$$$.

NoteIn the first test, there are three possible arrays $$$a$$$: $$$\{1\}$$$, $$$\{2\}$$$, and $$$ \{3\}$$$ and two possible arrays $$$b$$$: $$$\{\mathtt{L}\}$$$ and $$$\{\mathtt{R}\}$$$. For all six pairs of arrays $$$(a, b)$$$, the only participant will sit behind the computer $$$a_1$$$, so his madness will be $$$0$$$. So the total sum of madnesses will be $$$0$$$.In the second test, all possible pairs of arrays $$$(a, b)$$$, such that no person becomes upset are:  $$$(\{1, 1\}, \{\mathtt{L}, \mathtt{L}\})$$$, the sum of madnesses is $$$1$$$;  $$$(\{1, 1\}, \{\mathtt{R}, \mathtt{L}\})$$$, the sum of madnesses is $$$1$$$;  $$$(\{2, 2\}, \{\mathtt{R}, \mathtt{R}\})$$$, the sum of madnesses is $$$1$$$;  $$$(\{2, 2\}, \{\mathtt{L}, \mathtt{R}\})$$$, the sum of madnesses is $$$1$$$;  all possible pairs of $$$a \in \{\{1, 2\}, \{2, 1\}\}$$$ and $$$b \in \{\{\mathtt{L}, \mathtt{L}\}, \{\mathtt{R}, \mathtt{L}\}, \{\mathtt{L}, \mathtt{R}\}, \{\mathtt{R}, \mathtt{R}\}\}$$$, the sum of madnesses is $$$0$$$. So, the answer is $$$1 + 1 + 1 + 1 + 0 \ldots = 4$$$.


Let's call a graph with $$$n$$$ vertices, each of which has it's own point $$$A_i = (x_i, y_i)$$$ with integer coordinates, a planar tree if:  All points $$$A_1, A_2, \ldots, A_n$$$ are different and no three points lie on the same line.  The graph is a tree, i.e. there are exactly $$$n-1$$$ edges there exists a path between any pair of vertices.  For all pairs of edges $$$(s_1, f_1)$$$ and $$$(s_2, f_2)$$$, such that $$$s_1 \neq s_2, s_1 \neq f_2,$$$ $$$f_1 \neq s_2$$$, and $$$f_1 \neq f_2$$$, the segments $$$A_{s_1} A_{f_1}$$$ and $$$A_{s_2} A_{f_2}$$$ don't intersect. Imagine a planar tree with $$$n$$$ vertices. Consider the convex hull of points $$$A_1, A_2, \ldots, A_n$$$. Let's call this tree a spiderweb tree if for all $$$1 \leq i \leq n$$$ the following statements are true:  All leaves (vertices with degree $$$\leq 1$$$) of the tree lie on the border of the convex hull.  All vertices on the border of the convex hull are leaves. An example of a spiderweb tree:     The points $$$A_3, A_6, A_7, A_4$$$ lie on the convex hull and the leaf vertices of the tree are $$$3, 6, 7, 4$$$.Refer to the notes for more examples.Let's call the subset $$$S \subset \{1, 2, \ldots, n\}$$$ of vertices a subtree of the tree if for all pairs of vertices in $$$S$$$, there exists a path that contains only vertices from $$$S$$$. Note that any subtree of the planar tree is a planar tree.You are given a planar tree with $$$n$$$ vertexes. Let's call a partition of the set $$$\{1, 2, \ldots, n\}$$$ into non-empty subsets $$$A_1, A_2, \ldots, A_k$$$ (i.e. $$$A_i \cap A_j = \emptyset$$$ for all $$$1 \leq i < j \leq k$$$ and $$$A_1 \cup A_2 \cup \ldots \cup A_k = \{1, 2, \ldots, n\}$$$) good if for all $$$1 \leq i \leq k$$$, the subtree $$$A_i$$$ is a spiderweb tree. Two partitions are different if there exists some set that lies in one parition, but not the other.Find the number of good partitions. Since this number can be very large, find it modulo $$$998\,244\,353$$$.

The first line contains an integer $$$n$$$ ($$$1 \leq n \leq 100$$$) — the number of vertices in the tree. The next $$$n$$$ lines each contain two integers $$$x_i, y_i$$$ ($$$-10^9 \leq x_i, y_i \leq 10^9$$$)  — the coordinates of $$$i$$$-th vertex, $$$A_i$$$. The next $$$n-1$$$ lines contain two integers $$$s, f$$$ ($$$1 \leq s, f \leq n$$$) — the edges $$$(s, f)$$$ of the tree. It is guaranteed that all given points are different and that no three of them lie at the same line. Additionally, it is guaranteed that the given edges and coordinates of the points describe a planar tree.

Print one integer  — the number of good partitions of vertices of the given planar tree, modulo $$$998\,244\,353$$$.

Note    The picture for the first sample. In the first test case, all good partitions are:  $$$\{1\}$$$, $$$\{2\}$$$, $$$\{3\}$$$, $$$\{4\}$$$;  $$$\{1, 2\}$$$, $$$\{3\}$$$, $$$\{4\}$$$;  $$$\{1, 3\}$$$, $$$\{2\}$$$, $$$\{4\}$$$;  $$$\{1, 4\}$$$, $$$\{2\}$$$, $$$\{3\}$$$;  $$$\{1, 2, 3, 4\}$$$. The partition $$$\{1, 2, 3\}$$$, $$$\{4\}$$$ isn't good, because the subtree $$$\{1, 2, 3\}$$$ isn't spiderweb tree, since the non-leaf vertex $$$1$$$ lies on the convex hull.The partition $$$\{2, 3, 4\}$$$, $$$\{1\}$$$ isn't good, because the subset $$$\{2, 3, 4\}$$$ isn't a subtree.    The picture for the second sample. In the second test case, the given tree isn't a spiderweb tree, because the leaf vertex $$$1$$$ doesn't lie on the convex hull. However, the subtree $$$\{2, 3, 4, 5\}$$$ is a spiderweb tree.    The picture for the third sample.     The picture for the fourth sample. In the fourth test case, the partition $$$\{1, 2, 3, 4\}$$$, $$$\{5, 6, 7, 8\}$$$ is good because all subsets are spiderweb subtrees.


The circle line of the Roflanpolis subway has $$$n$$$ stations.There are two parallel routes in the subway. The first one visits stations in order $$$1 \to 2 \to \ldots \to n \to 1 \to 2 \to \ldots$$$ (so the next stop after station $$$x$$$ is equal to $$$(x+1)$$$ if $$$x < n$$$ and $$$1$$$ otherwise). The second route visits stations in order $$$n \to (n-1) \to \ldots \to 1 \to n \to (n-1) \to \ldots$$$ (so the next stop after station $$$x$$$ is equal to $$$(x-1)$$$ if $$$x>1$$$ and $$$n$$$ otherwise). All trains depart their stations simultaneously, and it takes exactly $$$1$$$ minute to arrive at the next station.Two toads live in this city, their names are Daniel and Vlad.Daniel is currently in a train of the first route at station $$$a$$$ and will exit the subway when his train reaches station $$$x$$$.Coincidentally, Vlad is currently in a train of the second route at station $$$b$$$ and he will exit the subway when his train reaches station $$$y$$$.Surprisingly, all numbers $$$a,x,b,y$$$ are distinct.Toad Ilya asks you to check if Daniel and Vlad will ever be at the same station at the same time during their journey. In other words, check if there is a moment when their trains stop at the same station. Note that this includes the moments when Daniel or Vlad enter or leave the subway.

The first line contains five space-separated integers $$$n$$$, $$$a$$$, $$$x$$$, $$$b$$$, $$$y$$$ ($$$4 \leq n \leq 100$$$, $$$1 \leq a, x, b, y \leq n$$$, all numbers among $$$a$$$, $$$x$$$, $$$b$$$, $$$y$$$ are distinct) — the number of stations in Roflanpolis, Daniel's start station, Daniel's finish station, Vlad's start station and Vlad's finish station, respectively.

Output "YES" if there is a time moment when Vlad and Daniel are at the same station, and "NO" otherwise. You can print each letter in any case (upper or lower).

NoteIn the first example, Daniel and Vlad start at the stations $$$(1, 3)$$$. One minute later they are at stations $$$(2, 2)$$$. They are at the same station at this moment. Note that Vlad leaves the subway right after that.Consider the second example, let's look at the stations Vlad and Daniel are at. They are:   initially $$$(2, 9)$$$,  after $$$1$$$ minute $$$(3, 8)$$$,  after $$$2$$$ minutes $$$(4, 7)$$$,  after $$$3$$$ minutes $$$(5, 6)$$$,  after $$$4$$$ minutes $$$(6, 5)$$$,  after $$$5$$$ minutes $$$(7, 4)$$$,  after $$$6$$$ minutes $$$(8, 3)$$$,  after $$$7$$$ minutes $$$(9, 2)$$$,  after $$$8$$$ minutes $$$(10, 1)$$$,  after $$$9$$$ minutes $$$(1, 10)$$$. After that, they both leave the subway because they are at their finish stations, so there is no moment when they both are at the same station.


Little penguin Polo loves his home village. The village has n houses, indexed by integers from 1 to n. Each house has a plaque containing an integer, the i-th house has a plaque containing integer pi (1 ≤ pi ≤ n).Little penguin Polo loves walking around this village. The walk looks like that. First he stands by a house number x. Then he goes to the house whose number is written on the plaque of house x (that is, to house px), then he goes to the house whose number is written on the plaque of house px (that is, to house ppx), and so on.We know that:  When the penguin starts walking from any house indexed from 1 to k, inclusive, he can walk to house number 1.  When the penguin starts walking from any house indexed from k + 1 to n, inclusive, he definitely cannot walk to house number 1.  When the penguin starts walking from house number 1, he can get back to house number 1 after some non-zero number of walks from a house to a house. You need to find the number of ways you may write the numbers on the houses' plaques so as to fulfill the three above described conditions. Print the remainder after dividing this number by 1000000007 (109 + 7).

The single line contains two space-separated integers n and k (1 ≤ n ≤ 1000, 1 ≤ k ≤ min(8, n)) — the number of the houses and the number k from the statement.

In a single line print a single integer — the answer to the problem modulo 1000000007 (109 + 7).


One day a bear lived on the Oxy axis. He was afraid of the dark, so he couldn't move at night along the plane points that aren't lit. One day the bear wanted to have a night walk from his house at point (l, 0) to his friend's house at point (r, 0), along the segment of length (r - l). Of course, if he wants to make this walk, he needs each point of the segment to be lit. That's why the bear called his friend (and yes, in the middle of the night) asking for a very delicate favor.The Oxy axis contains n floodlights. Floodlight i is at point (xi, yi) and can light any angle of the plane as large as ai degree with vertex at point (xi, yi). The bear asked his friend to turn the floodlights so that he (the bear) could go as far away from his house as possible during the walking along the segment. His kind friend agreed to fulfill his request. And while he is at it, the bear wonders: what is the furthest he can go away from his house? Hep him and find this distance.Consider that the plane has no obstacles and no other light sources besides the floodlights. The bear's friend cannot turn the floodlights during the bear's walk. Assume that after all the floodlights are turned in the correct direction, the bear goes for a walk and his friend goes to bed.

The first line contains three space-separated integers n, l, r (1 ≤ n ≤ 20;  - 105 ≤ l ≤ r ≤ 105). The i-th of the next n lines contain three space-separated integers xi, yi, ai ( - 1000 ≤ xi ≤ 1000; 1 ≤ yi ≤ 1000; 1 ≤ ai ≤ 90) — the floodlights' description.  Note that two floodlights can be at the same point of the plane.

Print a single real number — the answer to the problem. The answer will be considered correct if its relative or absolute error doesn't exceed 10 - 6.

NoteIn the first sample, one of the possible solutions is:   In the second sample, a single solution is:   In the third sample, a single solution is:


Lenny is playing a game on a 3 × 3 grid of lights. In the beginning of the game all lights are switched on. Pressing any of the lights will toggle it and all side-adjacent lights. The goal of the game is to switch all the lights off. We consider the toggling as follows: if the light was switched on then it will be switched off, if it was switched off then it will be switched on.Lenny has spent some time playing with the grid and by now he has pressed each light a certain number of times. Given the number of times each light is pressed, you have to print the current state of each light.

The input consists of three rows. Each row contains three integers each between 0 to 100 inclusive. The j-th number in the i-th row is the number of times the j-th light of the i-th row of the grid is pressed.

Print three lines, each containing three characters. The j-th character of the i-th line is "1" if and only if the corresponding light is switched on, otherwise it's "0".


In this task Anna and Maria play the following game. Initially they have a checkered piece of paper with a painted n × m rectangle (only the border, no filling). Anna and Maria move in turns and Anna starts. During each move one should paint inside the last-painted rectangle a new lesser rectangle (along the grid lines). The new rectangle should have no common points with the previous one. Note that when we paint a rectangle, we always paint only the border, the rectangles aren't filled.Nobody wins the game — Anna and Maria simply play until they have done k moves in total. Count the number of different ways to play this game.

The first and only line contains three integers: n, m, k (1 ≤ n, m, k ≤ 1000).

Print the single number — the number of the ways to play the game. As this number can be very big, print the value modulo 1000000007 (109 + 7).

NoteTwo ways to play the game are considered different if the final pictures are different. In other words, if one way contains a rectangle that is not contained in the other way.In the first sample Anna, who performs her first and only move, has only one possible action plan — insert a 1 × 1 square inside the given 3 × 3 square.In the second sample Anna has as much as 9 variants: 4 ways to paint a 1 × 1 square, 2 ways to insert a 1 × 2 rectangle vertically, 2 more ways to insert it horizontally and one more way is to insert a 2 × 2 square.


Smart Beaver decided to be not only smart, but also a healthy beaver! And so he began to attend physical education classes at school X. In this school, physical education has a very creative teacher. One of his favorite warm-up exercises is throwing balls. Students line up. Each one gets a single ball in the beginning. The balls are numbered from 1 to n (by the demand of the inventory commission).   Figure 1. The initial position for n = 5.  After receiving the balls the students perform the warm-up exercise. The exercise takes place in a few throws. For each throw the teacher chooses any two arbitrary different students who will participate in it. The selected students throw their balls to each other. Thus, after each throw the students remain in their positions, and the two balls are swapped.   Figure 2. The example of a throw.  In this case there was a throw between the students, who were holding the 2-nd and the 4-th balls. Since the warm-up has many exercises, each of them can only continue for little time. Therefore, for each student we know the maximum number of throws he can participate in. For this lessons maximum number of throws will be 1 or 2.Note that after all phases of the considered exercise any ball can end up with any student. Smart Beaver decided to formalize it and introduced the concept of the "ball order". The ball order is a sequence of n numbers that correspond to the order of balls in the line. The first number will match the number of the ball of the first from the left student in the line, the second number will match the ball of the second student, and so on. For example, in figure 2 the order of the balls was (1, 2, 3, 4, 5), and after the throw it was (1, 4, 3, 2, 5). Smart beaver knows the number of students and for each student he knows the maximum number of throws in which he can participate. And now he is wondering: what is the number of distinct ways of ball orders by the end of the exercise.

The first line contains a single number n — the number of students in the line and the number of balls. The next line contains exactly n space-separated integers. Each number corresponds to a student in the line (the i-th number corresponds to the i-th from the left student in the line) and shows the number of throws he can participate in. The input limits for scoring 30 points are (subproblem D1):    1 ≤ n ≤ 10.  The input limits for scoring 70 points are (subproblems D1+D2):    1 ≤ n ≤ 500.  The input limits for scoring 100 points are (subproblems D1+D2+D3):    1 ≤ n ≤ 1000000.

The output should contain a single integer — the number of variants of ball orders after the warm up exercise is complete. As the number can be rather large, print it modulo 1000000007 (109 + 7).


This is the easy version of the problem. The difference between the versions is in the constraints on the array elements. You can make hacks only if all versions of the problem are solved.You are given an array $$$[a_1, a_2, \dots, a_n]$$$. Your goal is to find the length of the longest subarray of this array such that the most frequent value in it is not unique. In other words, you are looking for a subarray such that if the most frequent value occurs $$$f$$$ times in this subarray, then at least $$$2$$$ different values should occur exactly $$$f$$$ times.An array $$$c$$$ is a subarray of an array $$$d$$$ if $$$c$$$ can be obtained from $$$d$$$ by deletion of several (possibly, zero or all) elements from the beginning and several (possibly, zero or all) elements from the end.

The first line contains a single integer $$$n$$$ ($$$1 \le n \le 200\,000$$$) — the length of the array. The second line contains $$$n$$$ integers $$$a_1, a_2, \ldots, a_n$$$ ($$$1 \le a_i \le min(n, 100)$$$) — elements of the array.

You should output exactly one integer  — the length of the longest subarray of the array whose most frequent value is not unique. If there is no such subarray, output $$$0$$$.

NoteIn the first sample, the subarray $$$[1, 1, 2, 2, 3, 3]$$$ is good, but $$$[1, 1, 2, 2, 3, 3, 3]$$$ isn't: in the latter there are $$$3$$$ occurrences of number $$$3$$$, and no other element appears $$$3$$$ times.


Polycarp takes part in a math show. He is given n tasks, each consists of k subtasks, numbered 1 through k. It takes him tj minutes to solve the j-th subtask of any task. Thus, time required to solve a subtask depends only on its index, but not on the task itself. Polycarp can solve subtasks in any order.By solving subtask of arbitrary problem he earns one point. Thus, the number of points for task is equal to the number of solved subtasks in it. Moreover, if Polycarp completely solves the task (solves all k of its subtasks), he recieves one extra point. Thus, total number of points he recieves for the complete solution of the task is k + 1.Polycarp has M minutes of time. What is the maximum number of points he can earn?

The first line contains three integer numbers n, k and M (1 ≤ n ≤ 45, 1 ≤ k ≤ 45, 0 ≤ M ≤ 2·109). The second line contains k integer numbers, values tj (1 ≤ tj ≤ 1000000), where tj is the time in minutes required to solve j-th subtask of any task.

Print the maximum amount of points Polycarp can earn in M minutes.

NoteIn the first example Polycarp can complete the first task and spend 1 + 2 + 3 + 4 = 10 minutes. He also has the time to solve one subtask of the second task in one minute.In the second example Polycarp can solve the first subtask of all five tasks and spend 5·1 = 5 minutes. Also he can solve the second subtasks of two tasks and spend 2·2 = 4 minutes. Thus, he earns 5 + 2 = 7 points in total.


Ania has a large integer $$$S$$$. Its decimal representation has length $$$n$$$ and doesn't contain any leading zeroes. Ania is allowed to change at most $$$k$$$ digits of $$$S$$$. She wants to do it in such a way that $$$S$$$ still won't contain any leading zeroes and it'll be minimal possible. What integer will Ania finish with?

The first line contains two integers $$$n$$$ and $$$k$$$ ($$$1 \leq n \leq 200\,000$$$, $$$0 \leq k \leq n$$$) — the number of digits in the decimal representation of $$$S$$$ and the maximum allowed number of changed digits. The second line contains the integer $$$S$$$. It's guaranteed that $$$S$$$ has exactly $$$n$$$ digits and doesn't contain any leading zeroes.

Output the minimal possible value of $$$S$$$ which Ania can end with. Note that the resulting integer should also have $$$n$$$ digits.

NoteA number has leading zeroes if it consists of at least two digits and its first digit is $$$0$$$. For example, numbers $$$00$$$, $$$00069$$$ and $$$0101$$$ have leading zeroes, while $$$0$$$, $$$3000$$$ and $$$1010$$$ don't have leading zeroes.


A plane contains a not necessarily convex polygon without self-intersections, consisting of n vertexes, numbered from 1 to n. There is a spider sitting on the border of the polygon, the spider can move like that:  Transfer. The spider moves from the point p1 with coordinates (x1, y1), lying on the polygon border, to the point p2 with coordinates (x2, y2), also lying on the border. The spider can't go beyond the polygon border as it transfers, that is, the spider's path from point p1 to point p2 goes along the polygon border. It's up to the spider to choose the direction of walking round the polygon border (clockwise or counterclockwise).  Descend. The spider moves from point p1 with coordinates (x1, y1) to point p2 with coordinates (x2, y2), at that points p1 and p2 must lie on one vertical straight line (x1 = x2), point p1 must be not lower than point p2 (y1 ≥ y2) and segment p1p2 mustn't have points, located strictly outside the polygon (specifically, the segment can have common points with the border). Initially the spider is located at the polygon vertex with number s. Find the length of the shortest path to the vertex number t, consisting of transfers and descends. The distance is determined by the usual Euclidean metric .

The first line contains integer n (3 ≤ n ≤ 105) — the number of vertexes of the given polygon. Next n lines contain two space-separated integers each — the coordinates of the polygon vertexes. The vertexes are listed in the counter-clockwise order. The coordinates of the polygon vertexes do not exceed 104 in their absolute value.  The last line contains two space-separated integers s and t (1 ≤ s, t ≤ n) — the start and the end vertexes of the sought shortest way.  Consider the polygon vertexes numbered in the order they are given in the input, that is, the coordinates of the first vertex are located on the second line of the input and the coordinates of the n-th vertex are on the (n + 1)-th line. It is guaranteed that the given polygon is simple, that is, it contains no self-intersections or self-tangencies.

In the output print a single real number — the length of the shortest way from vertex s to vertex t. The answer is considered correct, if its absolute or relative error does not exceed 10 - 6.

NoteIn the first sample the spider transfers along the side that connects vertexes 1 and 4.In the second sample the spider doesn't have to transfer anywhere, so the distance equals zero.In the third sample the best strategy for the spider is to transfer from vertex 3 to point (2,3), descend to point (2,1), and then transfer to vertex 1.


Small, but very brave, mouse Brain was not accepted to summer school of young villains. He was upset and decided to postpone his plans of taking over the world, but to become a photographer instead.As you may know, the coolest photos are on the film (because you can specify the hashtag #film for such).Brain took a lot of colourful pictures on colored and black-and-white film. Then he developed and translated it into a digital form. But now, color and black-and-white photos are in one folder, and to sort them, one needs to spend more than one hour!As soon as Brain is a photographer not programmer now, he asks you to help him determine for a single photo whether it is colored or black-and-white.Photo can be represented as a matrix sized n × m, and each element of the matrix stores a symbol indicating corresponding pixel color. There are only 6 colors:   'C' (cyan) 'M' (magenta) 'Y' (yellow) 'W' (white) 'G' (grey) 'B' (black) The photo is considered black-and-white if it has only white, black and grey pixels in it. If there are any of cyan, magenta or yellow pixels in the photo then it is considered colored.

The first line of the input contains two integers n and m (1 ≤ n, m ≤ 100) — the number of photo pixel matrix rows and columns respectively. Then n lines describing matrix rows follow. Each of them contains m space-separated characters describing colors of pixels in a row. Each character in the line is one of the 'C', 'M', 'Y', 'W', 'G' or 'B'.

Print the "#Black&White" (without quotes), if the photo is black-and-white and "#Color" (without quotes), if it is colored, in the only line.


In some country live wizards. They love playing with numbers. The blackboard has two numbers written on it — a and b. The order of the numbers is not important. Let's consider a ≤ b for the sake of definiteness. The players can cast one of the two spells in turns:  Replace b with b - ak. Number k can be chosen by the player, considering the limitations that k > 0 and b - ak ≥ 0. Number k is chosen independently each time an active player casts a spell.  Replace b with b mod a. If a > b, similar moves are possible.If at least one of the numbers equals zero, a player can't make a move, because taking a remainder modulo zero is considered somewhat uncivilized, and it is far too boring to subtract a zero. The player who cannot make a move, loses.To perform well in the magic totalizator, you need to learn to quickly determine which player wins, if both wizards play optimally: the one that moves first or the one that moves second.

The first line contains a single integer t — the number of input data sets (1 ≤ t ≤ 104). Each of the next t lines contains two integers a, b (0 ≤ a, b ≤ 1018). The numbers are separated by a space. Please do not use the %lld specificator to read or write 64-bit integers in С++. It is preferred to use the cin, cout streams or the %I64d specificator.

For any of the t input sets print "First" (without the quotes) if the player who moves first wins. Print "Second" (without the quotes) if the player who moves second wins. Print the answers to different data sets on different lines in the order in which they are given in the input.

NoteIn the first sample, the first player should go to (11,10). Then, after a single move of the second player to (1,10), he will take 10 modulo 1 and win.In the second sample the first player has two moves to (1,10) and (21,10). After both moves the second player can win.In the third sample, the first player has no moves.In the fourth sample, the first player wins in one move, taking 30 modulo 10.


At first, let's define function $$$f(x)$$$ as follows: $$$$$$ \begin{matrix} f(x) & = & \left\{ \begin{matrix} \frac{x}{2} & \mbox{if } x \text{ is even} \\ x - 1 & \mbox{otherwise } \end{matrix} \right. \end{matrix} $$$$$$We can see that if we choose some value $$$v$$$ and will apply function $$$f$$$ to it, then apply $$$f$$$ to $$$f(v)$$$, and so on, we'll eventually get $$$1$$$. Let's write down all values we get in this process in a list and denote this list as $$$path(v)$$$. For example, $$$path(1) = [1]$$$, $$$path(15) = [15, 14, 7, 6, 3, 2, 1]$$$, $$$path(32) = [32, 16, 8, 4, 2, 1]$$$.Let's write all lists $$$path(x)$$$ for every $$$x$$$ from $$$1$$$ to $$$n$$$. The question is next: what is the maximum value $$$y$$$ such that $$$y$$$ is contained in at least $$$k$$$ different lists $$$path(x)$$$?Formally speaking, you need to find maximum $$$y$$$ such that $$$\left| \{ x ~|~ 1 \le x \le n, y \in path(x) \} \right| \ge k$$$.

The first line contains two integers $$$n$$$ and $$$k$$$ ($$$1 \le k \le n \le 10^{18}$$$).

Print the only integer — the maximum value that is contained in at least $$$k$$$ paths.

NoteIn the first example, the answer is $$$5$$$, since $$$5$$$ occurs in $$$path(5)$$$, $$$path(10)$$$ and $$$path(11)$$$.In the second example, the answer is $$$4$$$, since $$$4$$$ occurs in $$$path(4)$$$, $$$path(5)$$$, $$$path(8)$$$, $$$path(9)$$$, $$$path(10)$$$ and $$$path(11)$$$.In the third example $$$n = k$$$, so the answer is $$$1$$$, since $$$1$$$ is the only number occuring in all paths for integers from $$$1$$$ to $$$20$$$.


Let's define the function $$$f$$$ of multiset $$$a$$$ as the multiset of number of occurences of every number, that is present in $$$a$$$.E.g., $$$f(\{5, 5, 1, 2, 5, 2, 3, 3, 9, 5\}) = \{1, 1, 2, 2, 4\}$$$.Let's define $$$f^k(a)$$$, as applying $$$f$$$ to array $$$a$$$ $$$k$$$ times: $$$f^k(a) = f(f^{k-1}(a)), f^0(a) = a$$$. E.g., $$$f^2(\{5, 5, 1, 2, 5, 2, 3, 3, 9, 5\}) = \{1, 2, 2\}$$$.You are given integers $$$n, k$$$ and you are asked how many different values the function $$$f^k(a)$$$ can have, where $$$a$$$ is arbitrary non-empty array with numbers of size no more than $$$n$$$. Print the answer modulo $$$998\,244\,353$$$.

The first and only line of input consists of two integers $$$n, k$$$ ($$$1 \le n, k \le 2020$$$).

Print one number — the number of different values of function $$$f^k(a)$$$ on all possible non-empty arrays with no more than $$$n$$$ elements modulo $$$998\,244\,353$$$.


For an array of integers $$$a$$$, let's define $$$|a|$$$ as the number of elements in it.Let's denote two functions: $$$F(a, k)$$$ is a function that takes an array of integers $$$a$$$ and a positive integer $$$k$$$. The result of this function is the array containing $$$|a|$$$ first elements of the array that you get by replacing each element of $$$a$$$ with exactly $$$k$$$ copies of that element.For example, $$$F([2, 2, 1, 3, 5, 6, 8], 2)$$$ is calculated as follows: first, you replace each element of the array with $$$2$$$ copies of it, so you obtain $$$[2, 2, 2, 2, 1, 1, 3, 3, 5, 5, 6, 6, 8, 8]$$$. Then, you take the first $$$7$$$ elements of the array you obtained, so the result of the function is $$$[2, 2, 2, 2, 1, 1, 3]$$$. $$$G(a, x, y)$$$ is a function that takes an array of integers $$$a$$$ and two different integers $$$x$$$ and $$$y$$$. The result of this function is the array $$$a$$$ with every element equal to $$$x$$$ replaced by $$$y$$$, and every element equal to $$$y$$$ replaced by $$$x$$$.For example, $$$G([1, 1, 2, 3, 5], 3, 1) = [3, 3, 2, 1, 5]$$$.An array $$$a$$$ is a parent of the array $$$b$$$ if:  either there exists a positive integer $$$k$$$ such that $$$F(a, k) = b$$$;  or there exist two different integers $$$x$$$ and $$$y$$$ such that $$$G(a, x, y) = b$$$. An array $$$a$$$ is an ancestor of the array $$$b$$$ if there exists a finite sequence of arrays $$$c_0, c_1, \dots, c_m$$$ ($$$m \ge 0$$$) such that $$$c_0$$$ is $$$a$$$, $$$c_m$$$ is $$$b$$$, and for every $$$i \in [1, m]$$$, $$$c_{i-1}$$$ is a parent of $$$c_i$$$.And now, the problem itself.You are given two integers $$$n$$$ and $$$k$$$. Your goal is to construct a sequence of arrays $$$s_1, s_2, \dots, s_m$$$ in such a way that:  every array $$$s_i$$$ contains exactly $$$n$$$ elements, and all elements are integers from $$$1$$$ to $$$k$$$;  for every array $$$a$$$ consisting of exactly $$$n$$$ integers from $$$1$$$ to $$$k$$$, the sequence contains at least one array $$$s_i$$$ such that $$$s_i$$$ is an ancestor of $$$a$$$. Print the minimum number of arrays in such sequence.

The only line contains two integers $$$n$$$ and $$$k$$$ ($$$1 \le n, k \le 2 \cdot 10^5$$$).

Print one integer — the minimum number of elements in a sequence of arrays meeting the constraints. Since the answer can be large, output it modulo $$$998244353$$$.


Ivan's classes at the university have just finished, and now he wants to go to the local CFK cafe and eat some fried chicken.CFK sells chicken chunks in small and large portions. A small portion contains 3 chunks; a large one — 7 chunks. Ivan wants to eat exactly x chunks. Now he wonders whether he can buy exactly this amount of chicken.Formally, Ivan wants to know if he can choose two non-negative integers a and b in such a way that a small portions and b large ones contain exactly x chunks.Help Ivan to answer this question for several values of x!

The first line contains one integer n (1 ≤ n ≤ 100) — the number of testcases. The i-th of the following n lines contains one integer xi (1 ≤ xi ≤ 100) — the number of chicken chunks Ivan wants to eat.

Print n lines, in i-th line output YES if Ivan can buy exactly xi chunks. Otherwise, print NO.

NoteIn the first example Ivan can buy two small portions.In the second example Ivan cannot buy exactly 5 chunks, since one small portion is not enough, but two small portions or one large is too much.


Getting closer and closer to a mathematician, Serval becomes a university student on math major in Japari University. On the Calculus class, his teacher taught him how to calculate the expected length of a random subsegment of a given segment. Then he left a bonus problem as homework, with the award of a garage kit from IOI. The bonus is to extend this problem to the general case as follows.You are given a segment with length $$$l$$$. We randomly choose $$$n$$$ segments by choosing two points (maybe with non-integer coordinates) from the given segment equiprobably and the interval between the two points forms a segment. You are given the number of random segments $$$n$$$, and another integer $$$k$$$. The $$$2n$$$ endpoints of the chosen segments split the segment into $$$(2n+1)$$$ intervals. Your task is to calculate the expected total length of those intervals that are covered by at least $$$k$$$ segments of the $$$n$$$ random segments.You should find the answer modulo $$$998244353$$$.

First line contains three space-separated positive integers $$$n$$$, $$$k$$$ and $$$l$$$ ($$$1\leq k \leq n \leq 2000$$$, $$$1\leq l\leq 10^9$$$).

Output one integer — the expected total length of all the intervals covered by at least $$$k$$$ segments of the $$$n$$$ random segments modulo $$$998244353$$$. Formally, let $$$M = 998244353$$$. It can be shown that the answer can be expressed as an irreducible fraction $$$\frac{p}{q}$$$, where $$$p$$$ and $$$q$$$ are integers and $$$q \not \equiv 0 \pmod{M}$$$. Output the integer equal to $$$p \cdot q^{-1} \bmod M$$$. In other words, output such an integer $$$x$$$ that $$$0 \le x < M$$$ and $$$x \cdot q \equiv p \pmod{M}$$$.

NoteIn the first example, the expected total length is $$$\int_0^1 \int_0^1 |x-y| \,\mathrm{d}x\,\mathrm{d}y = {1\over 3}$$$, and $$$3^{-1}$$$ modulo $$$998244353$$$ is $$$332748118$$$.


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:


Let's analyze a program written on some strange programming language. The variables in this language have names consisting of $$$1$$$ to $$$4$$$ characters, and each character is a lowercase or an uppercase Latin letter, or a digit. There is an extra constraint that the first character should not be a digit.There are four types of operations in the program, each denoted by one of the characters: $, ^, # or &.Each line of the program has one of the following formats:   <lvalue>=<rvalue>, where <lvalue> and <rvalue> are valid variable names;  <lvalue>=<arg1><op><arg2>, where <lvalue>, <arg1> and <arg2> are valid variable names, and <op> is an operation character. The program is executed line-by-line, and the result of execution is stored in a variable having the name res. If res is never assigned in the program, then the result will be equal to the value of res before running the program.Two programs are called equivalent if no matter which operations do characters $, ^, # and & denote (but, obviously, performing the same operation on the same arguments gives the same result) and which values do variables have before execution of program, the value of res after running the first program is equal to the value of res after running the second program (the programs are executed independently).You are given a program consisting of $$$n$$$ lines. Your task is to write a program consisting of minimum possible number of lines that is equivalent to the program you are given.

The first line contains one integer $$$n$$$ ($$$1 \le n \le 1000$$$) — the number of lines in the program. Then $$$n$$$ lines follow — the program itself. Each line corresponds to the format described in the statement and has no extra whitespaces.

In the first line print $$$k$$$ — the minimum number of lines in the equivalent program. Then print $$$k$$$ lines without any whitespaces — an equivalent program having exactly $$$k$$$ lines, in the same format it is described in the statement.


Sheldon, Leonard, Penny, Rajesh and Howard are in the queue for a "Double Cola" drink vending machine; there are no other people in the queue. The first one in the queue (Sheldon) buys a can, drinks it and doubles! The resulting two Sheldons go to the end of the queue. Then the next in the queue (Leonard) buys a can, drinks it and gets to the end of the queue as two Leonards, and so on. This process continues ad infinitum.For example, Penny drinks the third can of cola and the queue will look like this: Rajesh, Howard, Sheldon, Sheldon, Leonard, Leonard, Penny, Penny.Write a program that will print the name of a man who will drink the n-th can.Note that in the very beginning the queue looks like that: Sheldon, Leonard, Penny, Rajesh, Howard. The first person is Sheldon.

The input data consist of a single integer n (1 ≤ n ≤ 109). It is guaranteed that the pretests check the spelling of all the five names, that is, that they contain all the five possible answers.

Print the single line — the name of the person who drinks the n-th can of cola. The cans are numbered starting from 1. Please note that you should spell the names like this: "Sheldon", "Leonard", "Penny", "Rajesh", "Howard" (without the quotes). In that order precisely the friends are in the queue initially.


Dima loves representing an odd number as the sum of multiple primes, and Lisa loves it when there are at most three primes. Help them to represent the given number as the sum of at most than three primes.More formally, you are given an odd numer n. Find a set of numbers pi (1 ≤ i ≤ k), such that 1 ≤ k ≤ 3 pi is a prime The numbers pi do not necessarily have to be distinct. It is guaranteed that at least one possible solution exists.

The single line contains an odd number n (3 ≤ n < 109).

In the first line print k (1 ≤ k ≤ 3), showing how many numbers are in the representation you found. In the second line print numbers pi in any order. If there are multiple possible solutions, you can print any of them.

NoteA prime is an integer strictly larger than one that is divisible only by one and by itself.


It happened at the times of the Great Berland Empire. Once the Emperor dreamt that the Messenger from the gods ordered to build a temple whose base would be a convex polygon with n angles. Next morning the Emperor gave the command to build a temple whose base was a regular polygon with n angles. The temple was built but soon the Empire was shaken with disasters and crop failures. After an earthquake destroyed the temple, the Emperor understood that he somehow caused the wrath of gods to fall on his people. He ordered to bring the wise man. When the wise man appeared, the Emperor retold the dream to him and asked "Oh the wisest among the wisest, tell me how could I have infuriated the Gods?". "My Lord," the wise man answered. "As far as I can judge, the gods are angry because you were too haste to fulfill their order and didn't listen to the end of the message".Indeed, on the following night the Messenger appeared again. He reproached the Emperor for having chosen an imperfect shape for the temple. "But what shape can be more perfect than a regular polygon!?" cried the Emperor in his dream. To that the Messenger gave a complete and thorough reply.   All the vertices of the polygon should be positioned in the lattice points.  All the lengths of its sides should be different.  From the possible range of such polygons a polygon which maximum side is minimal possible must be chosen. You are an obedient architect who is going to make the temple's plan. Note that the polygon should be simple (having a border without self-intersections and overlapping) and convex, however, it is acceptable for three consecutive vertices to lie on the same line.

The first line contains the single number n (3 ≤ n ≤ 10000).

Print "YES" (without quotes) in the first line if it is possible to build a polygon possessing the needed qualities. In the next n lines print integer coordinates of the polygon vertices in the order in which they would be passed counter clockwise. The absolute value of the coordinates shouldn't exceed 109. No two vertices can coincide. It is permitted to print any of the possible solutions. Print "NO" if to build the polygon is impossible.


Sereja owns a restaurant for n people. The restaurant hall has a coat rack with n hooks. Each restaurant visitor can use a hook to hang his clothes on it. Using the i-th hook costs ai rubles. Only one person can hang clothes on one hook.Tonight Sereja expects m guests in the restaurant. Naturally, each guest wants to hang his clothes on an available hook with minimum price (if there are multiple such hooks, he chooses any of them). However if the moment a guest arrives the rack has no available hooks, Sereja must pay a d ruble fine to the guest. Help Sereja find out the profit in rubles (possibly negative) that he will get tonight. You can assume that before the guests arrive, all hooks on the rack are available, all guests come at different time, nobody besides the m guests is visiting Sereja's restaurant tonight.

The first line contains two integers n and d (1 ≤ n, d ≤ 100). The next line contains integers a1, a2, ..., an (1 ≤ ai ≤ 100). The third line contains integer m (1 ≤ m ≤ 100).

In a single line print a single integer — the answer to the problem.

NoteIn the first test both hooks will be used, so Sereja gets 1 + 2 = 3 rubles.In the second test both hooks will be used but Sereja pays a fine 8 times, so the answer is 3 - 8 =  - 5.


A number is called quasibinary if its decimal representation contains only digits 0 or 1. For example, numbers 0, 1, 101, 110011 — are quasibinary and numbers 2, 12, 900 are not.You are given a positive integer n. Represent it as a sum of minimum number of quasibinary numbers.

The first line contains a single integer n (1 ≤ n ≤ 106).

In the first line print a single integer k — the minimum number of numbers in the representation of number n as a sum of quasibinary numbers. In the second line print k numbers — the elements of the sum. All these numbers should be quasibinary according to the definition above, their sum should equal n. Do not have to print the leading zeroes in the numbers. The order of numbers doesn't matter. If there are multiple possible representations, you are allowed to print any of them.


Simon and Antisimon play a game. Initially each player receives one fixed positive integer that doesn't change throughout the game. Simon receives number a and Antisimon receives number b. They also have a heap of n stones. The players take turns to make a move and Simon starts. During a move a player should take from the heap the number of stones equal to the greatest common divisor of the fixed number he has received and the number of stones left in the heap. A player loses when he cannot take the required number of stones (i. e. the heap has strictly less stones left than one needs to take). Your task is to determine by the given a, b and n who wins the game.

The only string contains space-separated integers a, b and n (1 ≤ a, b, n ≤ 100) — the fixed numbers Simon and Antisimon have received correspondingly and the initial number of stones in the pile.

If Simon wins, print "0" (without the quotes), otherwise print "1" (without the quotes).

NoteThe greatest common divisor of two non-negative integers a and b is such maximum positive integer k, that a is divisible by k without remainder and similarly, b is divisible by k without remainder. Let gcd(a, b) represent the operation of calculating the greatest common divisor of numbers a and b. Specifically, gcd(x, 0) = gcd(0, x) = x.In the first sample the game will go like that: Simon should take gcd(3, 9) = 3 stones from the heap. After his move the heap has 6 stones left. Antisimon should take gcd(5, 6) = 1 stone from the heap. After his move the heap has 5 stones left. Simon should take gcd(3, 5) = 1 stone from the heap. After his move the heap has 4 stones left. Antisimon should take gcd(5, 4) = 1 stone from the heap. After his move the heap has 3 stones left. Simon should take gcd(3, 3) = 3 stones from the heap. After his move the heap has 0 stones left. Antisimon should take gcd(5, 0) = 5 stones from the heap. As 0 < 5, it is impossible and Antisimon loses.In the second sample each player during each move takes one stone from the heap. As n is even, Antisimon takes the last stone and Simon can't make a move after that.


DZY loves Fast Fourier Transformation, and he enjoys using it.Fast Fourier Transformation is an algorithm used to calculate convolution. Specifically, if a, b and c are sequences with length n, which are indexed from 0 to n - 1, andWe can calculate c fast using Fast Fourier Transformation.DZY made a little change on this formula. NowTo make things easier, a is a permutation of integers from 1 to n, and b is a sequence only containing 0 and 1. Given a and b, DZY needs your help to calculate c.Because he is naughty, DZY provides a special way to get a and b. What you need is only three integers n, d, x. After getting them, use the code below to generate a and b.//x is 64-bit variable;function getNextX() {    x = (x * 37 + 10007) % 1000000007;    return x;}function initAB() {    for(i = 0; i < n; i = i + 1){        a[i] = i + 1;    }    for(i = 0; i < n; i = i + 1){        swap(a[i], a[getNextX() % (i + 1)]);    }    for(i = 0; i < n; i = i + 1){        if (i < d)            b[i] = 1;        else            b[i] = 0;    }    for(i = 0; i < n; i = i + 1){        swap(b[i], b[getNextX() % (i + 1)]);    }}Operation x % y denotes remainder after division x by y. Function swap(x, y) swaps two values x and y.

The only line of input contains three space-separated integers n, d, x (1 ≤ d ≤ n ≤ 100000; 0 ≤ x ≤ 1000000006). Because DZY is naughty, x can't be equal to 27777500.

Output n lines, the i-th line should contain an integer ci - 1.

NoteIn the first sample, a is [1 3 2], b is [1 0 0], so c0 = max(1·1) = 1, c1 = max(1·0, 3·1) = 3, c2 = max(1·0, 3·0, 2·1) = 2.In the second sample, a is [2 1 4 5 3], b is [1 1 1 0 1].In the third sample, a is [5 2 1 4 3], b is [1 1 1 1 0].


After seeing the "ALL YOUR BASE ARE BELONG TO US" meme for the first time, numbers X and Y realised that they have different bases, which complicated their relations.You're given a number X represented in base bx and a number Y represented in base by. Compare those two numbers.

The first line of the input contains two space-separated integers n and bx (1 ≤ n ≤ 10, 2 ≤ bx ≤ 40), where n is the number of digits in the bx-based representation of X.  The second line contains n space-separated integers x1, x2, ..., xn (0 ≤ xi < bx) — the digits of X. They are given in the order from the most significant digit to the least significant one. The following two lines describe Y in the same way: the third line contains two space-separated integers m and by (1 ≤ m ≤ 10, 2 ≤ by ≤ 40, bx ≠ by), where m is the number of digits in the by-based representation of Y, and the fourth line contains m space-separated integers y1, y2, ..., ym (0 ≤ yi < by) — the digits of Y. There will be no leading zeroes. Both X and Y will be positive. All digits of both numbers are given in the standard decimal numeral system.

Output a single character (quotes for clarity):    '<' if X < Y  '>' if X > Y  '=' if X = Y

NoteIn the first sample, X = 1011112 = 4710 = Y.In the second sample, X = 1023 = 215 and Y = 245 = 1123, thus X < Y.In the third sample,  and Y = 48031509. We may notice that X starts with much larger digits and bx is much larger than by, so X is clearly larger than Y.


Gennady owns a small hotel in the countryside where he lives a peaceful life. He loves to take long walks, watch sunsets and play cards with tourists staying in his hotel. His favorite game is called "Mau-Mau".To play Mau-Mau, you need a pack of $$$52$$$ cards. Each card has a suit (Diamonds — D, Clubs — C, Spades — S, or Hearts — H), and a rank (2, 3, 4, 5, 6, 7, 8, 9, T, J, Q, K, or A).At the start of the game, there is one card on the table and you have five cards in your hand. You can play a card from your hand if and only if it has the same rank or the same suit as the card on the table.In order to check if you'd be a good playing partner, Gennady has prepared a task for you. Given the card on the table and five cards in your hand, check if you can play at least one card.

The first line of the input contains one string which describes the card on the table. The second line contains five strings which describe the cards in your hand. Each string is two characters long. The first character denotes the rank and belongs to the set $$$\{{\tt 2}, {\tt 3}, {\tt 4}, {\tt 5}, {\tt 6}, {\tt 7}, {\tt 8}, {\tt 9}, {\tt T}, {\tt J}, {\tt Q}, {\tt K}, {\tt A}\}$$$. The second character denotes the suit and belongs to the set $$$\{{\tt D}, {\tt C}, {\tt S}, {\tt H}\}$$$. All the cards in the input are different.

If it is possible to play a card from your hand, print one word "YES". Otherwise, print "NO". You can print each letter in any case (upper or lower).

NoteIn the first example, there is an Ace of Spades (AS) on the table. You can play an Ace of Diamonds (AD) because both of them are Aces.In the second example, you cannot play any card.In the third example, you can play an Ace of Diamonds (AD) because it has the same suit as a Four of Diamonds (4D), which lies on the table.


You are given a string $$$s$$$ of length $$$n$$$ consisting of lowercase Latin letters. You may apply some operations to this string: in one operation you can delete some contiguous substring of this string, if all letters in the substring you delete are equal. For example, after deleting substring bbbb from string abbbbaccdd we get the string aaccdd.Calculate the minimum number of operations to delete the whole string $$$s$$$.

The first line contains one integer $$$n$$$ ($$$1 \le n \le 500$$$) — the length of string $$$s$$$. The second line contains the string $$$s$$$ ($$$|s| = n$$$) consisting of lowercase Latin letters.

Output a single integer — the minimal number of operation to delete string $$$s$$$.


Polycarpus loves hamburgers very much. He especially adores the hamburgers he makes with his own hands. Polycarpus thinks that there are only three decent ingredients to make hamburgers from: a bread, sausage and cheese. He writes down the recipe of his favorite "Le Hamburger de Polycarpus" as a string of letters 'B' (bread), 'S' (sausage) и 'C' (cheese). The ingredients in the recipe go from bottom to top, for example, recipe "ВSCBS" represents the hamburger where the ingredients go from bottom to top as bread, sausage, cheese, bread and sausage again.Polycarpus has nb pieces of bread, ns pieces of sausage and nc pieces of cheese in the kitchen. Besides, the shop nearby has all three ingredients, the prices are pb rubles for a piece of bread, ps for a piece of sausage and pc for a piece of cheese.Polycarpus has r rubles and he is ready to shop on them. What maximum number of hamburgers can he cook? You can assume that Polycarpus cannot break or slice any of the pieces of bread, sausage or cheese. Besides, the shop has an unlimited number of pieces of each ingredient.

The first line of the input contains a non-empty string that describes the recipe of "Le Hamburger de Polycarpus". The length of the string doesn't exceed 100, the string contains only letters 'B' (uppercase English B), 'S' (uppercase English S) and 'C' (uppercase English C). The second line contains three integers nb, ns, nc (1 ≤ nb, ns, nc ≤ 100) — the number of the pieces of bread, sausage and cheese on Polycarpus' kitchen. The third line contains three integers pb, ps, pc (1 ≤ pb, ps, pc ≤ 100) — the price of one piece of bread, sausage and cheese in the shop. Finally, the fourth line contains integer r (1 ≤ r ≤ 1012) — the number of rubles Polycarpus has. Please, do not write the %lld specifier to read or write 64-bit integers in С++. It is preferred to use the cin, cout streams or the %I64d specifier.

Print the maximum number of hamburgers Polycarpus can make. If he can't make any hamburger, print 0.


There are n cards (n is even) in the deck. Each card has a positive integer written on it. n / 2 people will play new card game. At the beginning of the game each player gets two cards, each card is given to exactly one player. Find the way to distribute cards such that the sum of values written of the cards will be equal for each player. It is guaranteed that it is always possible.

The first line of the input contains integer n (2 ≤ n ≤ 100) — the number of cards in the deck. It is guaranteed that n is even. The second line contains the sequence of n positive integers a1, a2, ..., an (1 ≤ ai ≤ 100), where ai is equal to the number written on the i-th card.

Print n / 2 pairs of integers, the i-th pair denote the cards that should be given to the i-th player. Each card should be given to exactly one player. Cards are numbered in the order they appear in the input. It is guaranteed that solution exists. If there are several correct answers, you are allowed to print any of them.

NoteIn the first sample, cards are distributed in such a way that each player has the sum of numbers written on his cards equal to 8. In the second sample, all values ai are equal. Thus, any distribution is acceptable.


This version of the problem differs from the next one only in the constraint on $$$n$$$.Note that the memory limit in this problem is lower than in others.You have a vertical strip with $$$n$$$ cells, numbered consecutively from $$$1$$$ to $$$n$$$ from top to bottom.You also have a token that is initially placed in cell $$$n$$$. You will move the token up until it arrives at cell $$$1$$$.Let the token be in cell $$$x > 1$$$ at some moment. One shift of the token can have either of the following kinds:   Subtraction: you choose an integer $$$y$$$ between $$$1$$$ and $$$x-1$$$, inclusive, and move the token from cell $$$x$$$ to cell $$$x - y$$$.  Floored division: you choose an integer $$$z$$$ between $$$2$$$ and $$$x$$$, inclusive, and move the token from cell $$$x$$$ to cell $$$\lfloor \frac{x}{z} \rfloor$$$ ($$$x$$$ divided by $$$z$$$ rounded down). Find the number of ways to move the token from cell $$$n$$$ to cell $$$1$$$ using one or more shifts, and print it modulo $$$m$$$. Note that if there are several ways to move the token from one cell to another in one shift, all these ways are considered distinct (check example explanation for a better understanding).

The only line contains two integers $$$n$$$ and $$$m$$$ ($$$2 \le n \le 2 \cdot 10^5$$$; $$$10^8 < m < 10^9$$$; $$$m$$$ is a prime number) — the length of the strip and the modulo.

Print the number of ways to move the token from cell $$$n$$$ to cell $$$1$$$, modulo $$$m$$$.

NoteIn the first test, there are three ways to move the token from cell $$$3$$$ to cell $$$1$$$ in one shift: using subtraction of $$$y = 2$$$, or using division by $$$z = 2$$$ or $$$z = 3$$$.There are also two ways to move the token from cell $$$3$$$ to cell $$$1$$$ via cell $$$2$$$: first subtract $$$y = 1$$$, and then either subtract $$$y = 1$$$ again or divide by $$$z = 2$$$.Therefore, there are five ways in total.


Today an outstanding event is going to happen in the forest — hedgehog Filya will come to his old fried Sonya!Sonya is an owl and she sleeps during the day and stay awake from minute l1 to minute r1 inclusive. Also, during the minute k she prinks and is unavailable for Filya.Filya works a lot and he plans to visit Sonya from minute l2 to minute r2 inclusive.Calculate the number of minutes they will be able to spend together.

The only line of the input contains integers l1, r1, l2, r2 and k (1 ≤ l1, r1, l2, r2, k ≤ 1018, l1 ≤ r1, l2 ≤ r2), providing the segments of time for Sonya and Filya and the moment of time when Sonya prinks.

Print one integer — the number of minutes Sonya and Filya will be able to spend together.

NoteIn the first sample, they will be together during minutes 9 and 10.In the second sample, they will be together from minute 50 to minute 74 and from minute 76 to minute 100.


The stardate is 1977 and the science and art of detecting Death Stars is in its infancy. Princess Heidi has received information about the stars in the nearby solar system from the Rebel spies and now, to help her identify the exact location of the Death Star, she needs to know whether this information is correct. Two rebel spies have provided her with the maps of the solar system. Each map is an N × N grid, where each cell is either occupied by a star or empty. To see whether the information is correct, Heidi needs to know whether the two maps are of the same solar system, or if possibly one of the spies is actually an Empire double agent, feeding her false information.Unfortunately, spies may have accidentally rotated a map by 90, 180, or 270 degrees, or flipped it along the vertical or the horizontal axis, before delivering it to Heidi. If Heidi can rotate or flip the maps so that two of them become identical, then those maps are of the same solar system. Otherwise, there are traitors in the Rebel ranks! Help Heidi find out.

The first line of the input contains one number N (1 ≤ N ≤ 10) – the dimension of each map. Next N lines each contain N characters, depicting the first map: 'X' indicates a star, while 'O' indicates an empty quadrant of space. Next N lines each contain N characters, depicting the second map in the same format.

The only line of output should contain the word Yes if the maps are identical, or No if it is impossible to match them by performing rotations and translations.

4
XOOO
XXOO
OOOO
XXXX
XOOO
XOOO
XOXO
XOXX

NoteIn the first test, you can match the first map to the second map by first flipping the first map along the vertical axis, and then by rotating it 90 degrees clockwise.


Tavak and Seyyed are good friends. Seyyed is very funny and he told Tavak to solve the following problem instead of longest-path.You are given l and r. For all integers from l to r, inclusive, we wrote down all of their integer divisors except 1. Find the integer that we wrote down the maximum number of times.Solve the problem to show that it's not a NP problem.

The first line contains two integers l and r (2 ≤ l ≤ r ≤ 109).

Print single integer, the integer that appears maximum number of times in the divisors.  If there are multiple answers, print any of them.

NoteDefinition of a divisor: https://www.mathsisfun.com/definitions/divisor-of-an-integer-.htmlThe first example: from 19 to 29 these numbers are divisible by 2: {20, 22, 24, 26, 28}.The second example: from 3 to 6 these numbers are divisible by 3: {3, 6}.


Famous Brazil city Rio de Janeiro holds a tennis tournament and Ostap Bender doesn't want to miss this event. There will be n players participating, and the tournament will follow knockout rules from the very first game. That means, that if someone loses a game he leaves the tournament immediately.Organizers are still arranging tournament grid (i.e. the order games will happen and who is going to play with whom) but they have already fixed one rule: two players can play against each other only if the number of games one of them has already played differs by no more than one from the number of games the other one has already played. Of course, both players had to win all their games in order to continue participating in the tournament.Tournament hasn't started yet so the audience is a bit bored. Ostap decided to find out what is the maximum number of games the winner of the tournament can take part in (assuming the rule above is used). However, it is unlikely he can deal with this problem without your help.

The only line of the input contains a single integer n (2 ≤ n ≤ 1018) — the number of players to participate in the tournament.

Print the maximum number of games in which the winner of the tournament can take part.

NoteIn all samples we consider that player number 1 is the winner.In the first sample, there would be only one game so the answer is 1.In the second sample, player 1 can consequently beat players 2 and 3. In the third sample, player 1 can't play with each other player as after he plays with players 2 and 3 he can't play against player 4, as he has 0 games played, while player 1 already played 2. Thus, the answer is 2 and to achieve we make pairs (1, 2) and (3, 4) and then clash the winners.


Recall that a binary search tree is a rooted binary tree, whose nodes each store a key and each have at most two distinguished subtrees, left and right. The key in each node must be greater than any key stored in the left subtree, and less than any key stored in the right subtree.The depth of a vertex is the number of edges on the simple path from the vertex to the root. In particular, the depth of the root is $$$0$$$.Let's call a binary search tree perfectly balanced if there doesn't exist a binary search tree with the same number of vertices that has a strictly smaller sum of depths of its vertices.Let's call a binary search tree with integer keys striped if both of the following conditions are satisfied for every vertex $$$v$$$:   If $$$v$$$ has a left subtree whose root is $$$u$$$, then the parity of the key of $$$v$$$ is different from the parity of the key of $$$u$$$.  If $$$v$$$ has a right subtree whose root is $$$w$$$, then the parity of the key of $$$v$$$ is the same as the parity of the key of $$$w$$$. You are given a single integer $$$n$$$. Find the number of perfectly balanced striped binary search trees with $$$n$$$ vertices that have distinct integer keys between $$$1$$$ and $$$n$$$, inclusive. Output this number modulo $$$998\,244\,353$$$.

The only line contains a single integer $$$n$$$ ($$$1 \le n \le 10^6$$$), denoting the required number of vertices.

Output the number of perfectly balanced striped binary search trees with $$$n$$$ vertices and distinct integer keys between $$$1$$$ and $$$n$$$, inclusive, modulo $$$998\,244\,353$$$.

NoteIn the first example, this is the only tree that satisfies the conditions: In the second example, here are various trees that don't satisfy some condition:


Giga Tower is the tallest and deepest building in Cyberland. There are 17 777 777 777 floors, numbered from  - 8 888 888 888 to 8 888 888 888. In particular, there is floor 0 between floor  - 1 and floor 1. Every day, thousands of tourists come to this place to enjoy the wonderful view. In Cyberland, it is believed that the number "8" is a lucky number (that's why Giga Tower has 8 888 888 888 floors above the ground), and, an integer is lucky, if and only if its decimal notation contains at least one digit "8". For example, 8,  - 180, 808 are all lucky while 42,  - 10 are not. In the Giga Tower, if you write code at a floor with lucky floor number, good luck will always be with you (Well, this round is #278, also lucky, huh?).Tourist Henry goes to the tower to seek good luck. Now he is at the floor numbered a. He wants to find the minimum positive integer b, such that, if he walks b floors higher, he will arrive at a floor with a lucky number.

The only line of input contains an integer a ( - 109 ≤ a ≤ 109).

NoteFor the first sample, he has to arrive at the floor numbered 180.For the second sample, he will arrive at 8.Note that b should be positive, so the answer for the third sample is 10, not 0.


Nowadays, most of the internet advertisements are not statically linked to a web page. Instead, what will be shown to the person opening a web page is determined within 100 milliseconds after the web page is opened. Usually, multiple companies compete for each ad slot on the web page in an auction. Each of them receives a request with details about the user, web page and ad slot and they have to respond within those 100 milliseconds with a bid they would pay for putting an advertisement on that ad slot. The company that suggests the highest bid wins the auction and gets to place its advertisement. If there are several companies tied for the highest bid, the winner gets picked at random.However, the company that won the auction does not have to pay the exact amount of its bid. In most of the cases, a second-price auction is used. This means that the amount paid by the company is equal to the maximum of all the other bids placed for this ad slot.Let's consider one such bidding. There are n companies competing for placing an ad. The i-th of these companies will bid an integer number of microdollars equiprobably randomly chosen from the range between Li and Ri, inclusive. In the other words, the value of the i-th company bid can be any integer from the range [Li, Ri] with the same probability. Determine the expected value that the winner will have to pay in a second-price auction.

The first line of input contains an integer number n (2 ≤ n ≤ 5). n lines follow, the i-th of them containing two numbers Li and Ri (1 ≤ Li ≤ Ri ≤ 10000) describing the i-th company's bid preferences. This problem doesn't have subproblems. You will get 8 points for the correct submission.

Output the answer with absolute or relative error no more than 1e - 9.

NoteConsider the first example. The first company bids a random integer number of microdollars in range [4, 7]; the second company bids between 8 and 10, and the third company bids 5 microdollars. The second company will win regardless of the exact value it bids, however the price it will pay depends on the value of first company's bid. With probability 0.5 the first company will bid at most 5 microdollars, and the second-highest price of the whole auction will be 5. With probability 0.25 it will bid 6 microdollars, and with probability 0.25 it will bid 7 microdollars. Thus, the expected value the second company will have to pay is 0.5·5 + 0.25·6 + 0.25·7 = 5.75.


Polycarp knows that if the sum of the digits of a number is divisible by $$$3$$$, then the number itself is divisible by $$$3$$$. He assumes that the numbers, the sum of the digits of which is divisible by $$$4$$$, are also somewhat interesting. Thus, he considers a positive integer $$$n$$$ interesting if its sum of digits is divisible by $$$4$$$.Help Polycarp find the nearest larger or equal interesting number for the given number $$$a$$$. That is, find the interesting number $$$n$$$ such that $$$n \ge a$$$ and $$$n$$$ is minimal.

The only line in the input contains an integer $$$a$$$ ($$$1 \le a \le 1000$$$).

Print the nearest greater or equal interesting number for the given number $$$a$$$. In other words, print the interesting number $$$n$$$ such that $$$n \ge a$$$ and $$$n$$$ is minimal.


Alice and Bob begin their day with a quick game. They first choose a starting number X0 ≥ 3 and try to reach one million by the process described below. Alice goes first and then they take alternating turns. In the i-th turn, the player whose turn it is selects a prime number smaller than the current number, and announces the smallest multiple of this prime number that is not smaller than the current number.Formally, he or she selects a prime p < Xi - 1 and then finds the minimum Xi ≥ Xi - 1 such that p divides Xi. Note that if the selected prime p already divides Xi - 1, then the number does not change.Eve has witnessed the state of the game after two turns. Given X2, help her determine what is the smallest possible starting number X0. Note that the players don't necessarily play optimally. You should consider all possible game evolutions.

The input contains a single integer X2 (4 ≤ X2 ≤ 106). It is guaranteed that the integer X2 is composite, that is, is not prime.

Output a single integer — the minimum possible X0.

NoteIn the first test, the smallest possible starting number is X0 = 6. One possible course of the game is as follows:   Alice picks prime 5 and announces X1 = 10  Bob picks prime 7 and announces X2 = 14. In the second case, let X0 = 15.   Alice picks prime 2 and announces X1 = 16  Bob picks prime 5 and announces X2 = 20.


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.


Unary is a minimalistic Brainfuck dialect in which programs are written using only one token. Brainfuck programs use 8 commands: "+", "-", "[", "]", "<", ">", "." and "," (their meaning is not important for the purposes of this problem). Unary programs are created from Brainfuck programs using the following algorithm. First, replace each command with a corresponding binary code, using the following conversion table:   ">"  →  1000,  "<"  →  1001,  "+"  →  1010,  "-"  →  1011,  "."  →  1100,  ","  →  1101,  "["  →  1110,  "]"  →  1111. Next, concatenate the resulting binary codes into one binary number in the same order as in the program. Finally, write this number using unary numeral system — this is the Unary program equivalent to the original Brainfuck one.You are given a Brainfuck program. Your task is to calculate the size of the equivalent Unary program, and print it modulo 1000003 (106 + 3).

The input will consist of a single line p which gives a Brainfuck program. String p will contain between 1 and 100 characters, inclusive. Each character of p will be "+", "-", "[", "]", "<", ">", "." or ",".

Output the size of the equivalent Unary program modulo 1000003 (106 + 3).

NoteTo write a number n in unary numeral system, one simply has to write 1 n times. For example, 5 written in unary system will be 11111.In the first example replacing Brainfuck commands with binary code will give us 1101 1100. After we concatenate the codes, we'll get 11011100 in binary system, or 220 in decimal. That's exactly the number of tokens in the equivalent Unary program.