Alice and Bob decided to eat some fruit. In the kitchen they found a large bag of oranges and apples. Alice immediately took an orange for herself, Bob took an apple. To make the process of sharing the remaining fruit more fun, the friends decided to play a game. They put multiple cards and on each one they wrote a letter, either 'A', or the letter 'B'. Then they began to remove the cards one by one from left to right, every time they removed a card with the letter 'A', Alice gave Bob all the fruits she had at that moment and took out of the bag as many apples and as many oranges as she had before. Thus the number of oranges and apples Alice had, did not change. If the card had written letter 'B', then Bob did the same, that is, he gave Alice all the fruit that he had, and took from the bag the same set of fruit. After the last card way removed, all the fruit in the bag were over.You know how many oranges and apples was in the bag at first. Your task is to find any sequence of cards that Alice and Bob could have played with.

The first line of the input contains two integers, x, y (1 ≤ x, y ≤ 1018, xy > 1) — the number of oranges and apples that were initially in the bag.

Print any sequence of cards that would meet the problem conditions as a compressed string of characters 'A' and 'B. That means that you need to replace the segments of identical consecutive characters by the number of repetitions of the characters and the actual character. For example, string AAABAABBB should be replaced by string 3A1B2A3B, but cannot be replaced by 2A1A1B2A3B or by 3AB2A3B. See the samples for clarifications of the output format. The string that you print should consist of at most 106 characters. It is guaranteed that if the answer exists, its compressed representation exists, consisting of at most 106 characters. If there are several possible answers, you are allowed to print any of them. If the sequence of cards that meet the problem statement does not not exist, print a single word Impossible.

NoteIn the first sample, if the row contained three cards with letter 'B', then Bob should give one apple to Alice three times. So, in the end of the game Alice has one orange and three apples, and Bob has one apple, in total it is one orange and four apples.In second sample, there is no answer since one card is not enough for game to finish, and two cards will produce at least three apples or three oranges.In the third sample, cards contain letters 'AB', so after removing the first card Bob has one orange and one apple, and after removal of second card Alice has two oranges and one apple. So, in total it is three oranges and two apples.


In the game Lizard Era: Beginning the protagonist will travel with three companions: Lynn, Meliana and Worrigan. Overall the game has n mandatory quests. To perform each of them, you need to take exactly two companions.The attitude of each of the companions to the hero is an integer. Initially, the attitude of each of them to the hero of neutral and equal to 0. As the hero completes quests, he makes actions that change the attitude of the companions, whom he took to perform this task, in positive or negative direction.Tell us what companions the hero needs to choose to make their attitude equal after completing all the quests. If this can be done in several ways, choose the one in which the value of resulting attitude is greatest possible.

The first line contains positive integer n (1 ≤ n ≤ 25) — the number of important tasks.  Next n lines contain the descriptions of the tasks — the i-th line contains three integers li, mi, wi — the values by which the attitude of Lynn, Meliana and Worrigan respectively will change towards the hero if the hero takes them on the i-th task. All the numbers in the input are integers and do not exceed 107 in absolute value.

If there is no solution, print in the first line "Impossible". Otherwise, print n lines, two characters is each line — in the i-th line print the first letters of the companions' names that hero should take to complete the i-th task ('L' for Lynn, 'M' for Meliana, 'W' for Worrigan). Print the letters in any order, if there are multiple solutions, print any of them.


The mobile application store has a new game called "Subway Roller".The protagonist of the game Philip is located in one end of the tunnel and wants to get out of the other one. The tunnel is a rectangular field consisting of three rows and n columns. At the beginning of the game the hero is in some cell of the leftmost column. Some number of trains rides towards the hero. Each train consists of two or more neighbouring cells in some row of the field.All trains are moving from right to left at a speed of two cells per second, and the hero runs from left to right at the speed of one cell per second. For simplicity, the game is implemented so that the hero and the trains move in turns. First, the hero moves one cell to the right, then one square up or down, or stays idle. Then all the trains move twice simultaneously one cell to the left. Thus, in one move, Philip definitely makes a move to the right and can move up or down. If at any point, Philip is in the same cell with a train, he loses. If the train reaches the left column, it continues to move as before, leaving the tunnel.Your task is to answer the question whether there is a sequence of movements of Philip, such that he would be able to get to the rightmost column.

Each test contains from one to ten sets of the input data. The first line of the test contains a single integer t (1 ≤ t ≤ 10 for pretests and tests or t = 1 for hacks; see the Notes section for details) — the number of sets. Then follows the description of t sets of the input data.  The first line of the description of each set contains two integers n, k (2 ≤ n ≤ 100, 1 ≤ k ≤ 26) — the number of columns on the field and the number of trains. Each of the following three lines contains the sequence of n character, representing the row of the field where the game is on. Philip's initial position is marked as 's', he is in the leftmost column. Each of the k trains is marked by some sequence of identical uppercase letters of the English alphabet, located in one line. Distinct trains are represented by distinct letters. Character '.' represents an empty cell, that is, the cell that doesn't contain either Philip or the trains.

For each set of the input data print on a single line word YES, if it is possible to win the game and word NO otherwise.

NoteIn the first set of the input of the first sample Philip must first go forward and go down to the third row of the field, then go only forward, then go forward and climb to the second row, go forward again and go up to the first row. After that way no train blocks Philip's path, so he can go straight to the end of the tunnel.Note that in this problem the challenges are restricted to tests that contain only one testset.


Vitalik the philatelist has a birthday today!As he is a regular customer in a stamp store called 'Robin Bobin', the store management decided to make him a gift.Vitalik wants to buy one stamp and the store will give him a non-empty set of the remaining stamps, such that the greatest common divisor (GCD) of the price of the stamps they give to him is more than one. If the GCD of prices of the purchased stamp and prices of present stamps set will be equal to 1, then Vitalik will leave the store completely happy.The store management asks you to count the number of different situations in which Vitalik will leave the store completely happy. Since the required number of situations can be very large, you need to find the remainder of this number modulo 109 + 7. The situations are different if the stamps purchased by Vitalik are different, or if one of the present sets contains a stamp that the other present does not contain.

The first line of the input contains integer n (2 ≤ n ≤ 5·105) — the number of distinct stamps, available for sale in the 'Robin Bobin' store.  The second line contains a sequence of integers a1, a2, ..., an (2 ≤ ai ≤ 107), where ai is the price of the i-th stamp.

Print a single integer — the remainder of the sought number of situations modulo 109 + 7.

NoteIn the first sample the following situations are possible:   Vitalik buys the 1-st stamp, the store gives him the 2-nd stamp as a present;  Vitalik buys the 3-rd stamp, the store gives him the 2-nd stamp as a present;  Vitalik buys the 2-nd stamp, the store gives him the 1-st stamp as a present;  Vitalik buys the 2-nd stamp, the store gives him the 3-rd stamp as a present;  Vitalik buys the 2-nd stamp, the store gives him the 1-st and 3-rd stamps as a present.


Given a sequence of integers a1, ..., an and q queries x1, ..., xq on it. For each query xi you have to count the number of pairs (l, r) such that 1 ≤ l ≤ r ≤ n and gcd(al, al + 1, ..., ar) = xi. is a greatest common divisor of v1, v2, ..., vn, that is equal to a largest positive integer that divides all vi.

The first line of the input contains integer n, (1 ≤ n ≤ 105), denoting the length of the sequence. The next line contains n space separated integers a1, ..., an, (1 ≤ ai ≤ 109). The third line of the input contains integer q, (1 ≤ q ≤ 3 × 105), denoting the number of queries. Then follows q lines, each contain an integer xi, (1 ≤ xi ≤ 109).

For each query print the result in a separate line.


Alena has successfully passed the entrance exams to the university and is now looking forward to start studying.One two-hour lesson at the Russian university is traditionally called a pair, it lasts for two academic hours (an academic hour is equal to 45 minutes).The University works in such a way that every day it holds exactly n lessons. Depending on the schedule of a particular group of students, on a given day, some pairs may actually contain classes, but some may be empty (such pairs are called breaks).The official website of the university has already published the schedule for tomorrow for Alena's group. Thus, for each of the n pairs she knows if there will be a class at that time or not.Alena's House is far from the university, so if there are breaks, she doesn't always go home. Alena has time to go home only if the break consists of at least two free pairs in a row, otherwise she waits for the next pair at the university.Of course, Alena does not want to be sleepy during pairs, so she will sleep as long as possible, and will only come to the first pair that is presented in her schedule. Similarly, if there are no more pairs, then Alena immediately goes home.Alena appreciates the time spent at home, so she always goes home when it is possible, and returns to the university only at the beginning of the next pair. Help Alena determine for how many pairs she will stay at the university. Note that during some pairs Alena may be at the university waiting for the upcoming pair.

The first line of the input contains a positive integer n (1 ≤ n ≤ 100) — the number of lessons at the university.  The second line contains n numbers ai (0 ≤ ai ≤ 1). Number ai equals 0, if Alena doesn't have the i-th pairs, otherwise it is equal to 1. Numbers a1, a2, ..., an are separated by spaces.

Print a single number — the number of pairs during which Alena stays at the university.

NoteIn the first sample Alena stays at the university from the second to the fifth pair, inclusive, during the third pair she will be it the university waiting for the next pair. In the last sample Alena doesn't have a single pair, so she spends all the time at home.


A little boy Laurenty has been playing his favourite game Nota for quite a while and is now very hungry. The boy wants to make sausage and cheese sandwiches, but first, he needs to buy a sausage and some cheese.The town where Laurenty lives in is not large. The houses in it are located in two rows, n houses in each row. Laurenty lives in the very last house of the second row. The only shop in town is placed in the first house of the first row.The first and second rows are separated with the main avenue of the city. The adjacent houses of one row are separated by streets.Each crosswalk of a street or an avenue has some traffic lights. In order to cross the street, you need to press a button on the traffic light, wait for a while for the green light and cross the street. Different traffic lights can have different waiting time.The traffic light on the crosswalk from the j-th house of the i-th row to the (j + 1)-th house of the same row has waiting time equal to aij (1 ≤ i ≤ 2, 1 ≤ j ≤ n - 1). For the traffic light on the crossing from the j-th house of one row to the j-th house of another row the waiting time equals bj (1 ≤ j ≤ n). The city doesn't have any other crossings.The boy wants to get to the store, buy the products and go back. The main avenue of the city is wide enough, so the boy wants to cross it exactly once on the way to the store and exactly once on the way back home. The boy would get bored if he had to walk the same way again, so he wants the way home to be different from the way to the store in at least one crossing.    Figure to the first sample. Help Laurenty determine the minimum total time he needs to wait at the crossroads.

The first line of the input contains integer n (2 ≤ n ≤ 50) — the number of houses in each row.  Each of the next two lines contains n - 1 space-separated integer — values aij (1 ≤ aij ≤ 100).  The last line contains n space-separated integers bj (1 ≤ bj ≤ 100).

Print a single integer — the least total time Laurenty needs to wait at the crossroads, given that he crosses the avenue only once both on his way to the store and on his way back home.

NoteThe first sample is shown on the figure above. In the second sample, Laurenty's path can look as follows:   Laurenty crosses the avenue, the waiting time is 3;  Laurenty uses the second crossing in the first row, the waiting time is 2;  Laurenty uses the first crossing in the first row, the waiting time is 1;  Laurenty uses the first crossing in the first row, the waiting time is 1;  Laurenty crosses the avenue, the waiting time is 1;  Laurenty uses the second crossing in the second row, the waiting time is 3.  In total we get that the answer equals 11.In the last sample Laurenty visits all the crossings, so the answer is 4.


Vasily has recently learned about the amazing properties of number π. In one of the articles it has been hypothesized that, whatever the sequence of numbers we have, in some position, this sequence is found among the digits of number π. Thus, if you take, for example, the epic novel "War and Peace" of famous Russian author Leo Tolstoy, and encode it with numbers, then we will find the novel among the characters of number π.Vasily was absolutely delighted with this, because it means that all the books, songs and programs have already been written and encoded in the digits of π. Vasily is, of course, a bit wary that this is only a hypothesis and it hasn't been proved, so he decided to check it out.To do this, Vasily downloaded from the Internet the archive with the sequence of digits of number π, starting with a certain position, and began to check the different strings of digits on the presence in the downloaded archive. Vasily quickly found short strings of digits, but each time he took a longer string, it turned out that it is not in the archive. Vasily came up with a definition that a string of length d is a half-occurrence if it contains a substring of length of at least , which occurs in the archive.To complete the investigation, Vasily took 2 large numbers x, y (x ≤ y) with the same number of digits and now he wants to find the number of numbers in the interval from x to y, which are half-occurrences in the archive. Help Vasily calculate this value modulo 109 + 7.

The first line contains string s consisting of decimal digits (1 ≤ |s| ≤ 1000) that Vasily will use to search substrings in. According to hypothesis, this sequence of digis indeed occurs in the decimal representation of π, although we can't guarantee that. The second and third lines contain two positive integers x, y of the same length d (x ≤ y, 2 ≤ d ≤ 50). Numbers x, y do not contain leading zeroes.

Print how many numbers in the segment from x to y that are half-occurrences in s modulo 109 + 7.


While Duff was resting in the beach, she accidentally found a strange array b0, b1, ..., bl - 1 consisting of l positive integers. This array was strange because it was extremely long, but there was another (maybe shorter) array, a0, ..., an - 1 that b can be build from a with formula: bi = ai mod n where a mod b denoted the remainder of dividing a by b.  Duff is so curious, she wants to know the number of subsequences of b like bi1, bi2, ..., bix (0 ≤ i1 < i2 < ... < ix < l), such that:   1 ≤ x ≤ k  For each 1 ≤ j ≤ x - 1,   For each 1 ≤ j ≤ x - 1, bij ≤ bij + 1. i.e this subsequence is non-decreasing. Since this number can be very large, she want to know it modulo 109 + 7.Duff is not a programmer, and Malek is unavailable at the moment. So she asked for your help. Please tell her this number.

The first line of input contains three integers, n, l and k (1 ≤ n, k, n × k ≤ 106 and 1 ≤ l ≤ 1018). The second line contains n space separated integers, a0, a1, ..., an - 1 (1 ≤ ai ≤ 109 for each 0 ≤ i ≤ n - 1).

Print the answer modulo 1 000 000 007 in one line.

NoteIn the first sample case, . So all such sequences are: , , , , , , , ,  and .


Recently Duff has been a soldier in the army. Malek is her commander.Their country, Andarz Gu has n cities (numbered from 1 to n) and n - 1 bidirectional roads. Each road connects two different cities. There exist a unique path between any two cities.There are also m people living in Andarz Gu (numbered from 1 to m). Each person has and ID number. ID number of i - th person is i and he/she lives in city number ci. Note that there may be more than one person in a city, also there may be no people living in the city.  Malek loves to order. That's why he asks Duff to answer to q queries. In each query, he gives her numbers v, u and a.To answer a query:Assume there are x people living in the cities lying on the path from city v to city u. Assume these people's IDs are p1, p2, ..., px in increasing order. If k = min(x, a), then Duff should tell Malek numbers k, p1, p2, ..., pk in this order. In the other words, Malek wants to know a minimums on that path (or less, if there are less than a people).Duff is very busy at the moment, so she asked you to help her and answer the queries.

The first line of input contains three integers, n, m and q (1 ≤ n, m, q ≤ 105). The next n - 1 lines contain the roads. Each line contains two integers v and u, endpoints of a road (1 ≤ v, u ≤ n, v ≠ u). Next line contains m integers c1, c2, ..., cm separated by spaces (1 ≤ ci ≤ n for each 1 ≤ i ≤ m). Next q lines contain the queries. Each of them contains three integers, v, u and a (1 ≤ v, u ≤ n and 1 ≤ a ≤ 10).

For each query, print numbers k, p1, p2, ..., pk separated by spaces in one line.

NoteGraph of Andarz Gu in the sample case is as follows (ID of people in each city are written next to them):


Duff is one if the heads of Mafia in her country, Andarz Gu. Andarz Gu has n cities (numbered from 1 to n) connected by m bidirectional roads (numbered by 1 to m).Each road has a destructing time, and a color. i-th road connects cities vi and ui and its color is ci and its destructing time is ti.Mafia wants to destruct a matching in Andarz Gu. A matching is a subset of roads such that no two roads in this subset has common endpoint. They can destruct these roads in parallel, i. e. the total destruction time is a maximum over destruction times of all selected roads.  They want two conditions to be satisfied:  The remaining roads form a proper coloring.  Destructing time of this matching is minimized. The remaining roads after destructing this matching form a proper coloring if and only if no two roads of the same color have same endpoint, or, in the other words, edges of each color should form a matching.There is no programmer in Mafia. That's why Duff asked for your help. Please help her and determine which matching to destruct in order to satisfied those conditions (or state that this is not possible).

The first line of input contains two integers n and m (2 ≤ n ≤ 5 × 104 and 1 ≤ m ≤ 5 × 104), number of cities and number of roads in the country. The next m lines contain the the roads. i - th of them contains four integers vi, ui, ci and ti (1 ≤ vi, ui ≤ n, vi ≠ ui and 1 ≤ ci, ti ≤ 109 for each 1 ≤ i ≤ m).

In the first line of input, print "Yes" (without quotes) if satisfying the first condition is possible and "No" (without quotes) otherwise. If it is possible, then you have to print two integers t and k in the second line, the minimum destructing time and the number of roads in the matching (). In the third line print k distinct integers separated by spaces, indices of the roads in the matching in any order. Roads are numbered starting from one in order of their appearance in the input. If there's more than one solution, print any of them.

NoteGraph of Andarz Gu in the first sample case is as follows:  A solution would be to destruct the roads with crosses.Graph of Andarz Gu in the second sample case is as follows:


Recently, Duff has been practicing weight lifting. As a hard practice, Malek gave her a task. He gave her a sequence of weights. Weight of i-th of them is 2wi pounds. In each step, Duff can lift some of the remaining weights and throw them away. She does this until there's no more weight left. Malek asked her to minimize the number of steps.  Duff is a competitive programming fan. That's why in each step, she can only lift and throw away a sequence of weights 2a1, ..., 2ak if and only if there exists a non-negative integer x such that 2a1 + 2a2 + ... + 2ak = 2x, i. e. the sum of those numbers is a power of two.Duff is a competitive programming fan, but not a programmer. That's why she asked for your help. Help her minimize the number of steps.

The first line of input contains integer n (1 ≤ n ≤ 106), the number of weights. The second line contains n integers w1, ..., wn separated by spaces (0 ≤ wi ≤ 106 for each 1 ≤ i ≤ n), the powers of two forming the weights values.

Print the minimum number of steps in a single line.

NoteIn the first sample case: One optimal way would be to throw away the first three in the first step and the rest in the second step. Also, it's not possible to do it in one step because their sum is not a power of two.In the second sample case: The only optimal way is to throw away one weight in each step. It's not possible to do it in less than 4 steps because there's no subset of weights with more than one weight and sum equal to a power of two.


Duff is the queen of her country, Andarz Gu. She's a competitive programming fan. That's why, when he saw her minister, Malek, free, she gave her a sequence consisting of n non-negative integers, a1, a2, ..., an and asked him to perform q queries for her on this sequence.  There are two types of queries:  given numbers l, r and k, Malek should perform  for each l ≤ i ≤ r (, bitwise exclusive OR of numbers a and b).  given numbers l and r Malek should tell her the score of sequence al, al + 1, ... , ar. Score of a sequence b1, ..., bk is the number of its different Kheshtaks. A non-negative integer w is a Kheshtak of this sequence if and only if there exists a subsequence of b, let's denote it as bi1, bi2, ... , bix (possibly empty) such that  (1 ≤ i1 < i2 < ... < ix ≤ k). If this subsequence is empty, then w = 0.Unlike Duff, Malek is not a programmer. That's why he asked for your help. Please help him perform these queries.

The first line of input contains two integers, n and q (1 ≤ n ≤ 2 × 105 and 1 ≤ q ≤ 4 × 104). The second line of input contains n integers, a1, a2, ..., an separated by spaces (0 ≤ ai ≤ 109 for each 1 ≤ i ≤ n). The next q lines contain the queries. Each line starts with an integer t (1 ≤ t ≤ 2), type of the corresponding query. If t = 1, then there are three more integers in that line, l, r and k. Otherwise there are two more integers, l and r. (1 ≤ l ≤ r ≤ n and 0 ≤ k ≤ 109)

Print the answer of each query of the second type in one line.

NoteIn the first query, we want all Kheshtaks of sequence 1, 2, 3, 4, 2 which are: 0, 1, 2, 3, 4, 5, 6, 7.In the third query, we want all Khestaks of sequence 1, 10, 3, 4, 2 which are: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15.In the fifth query, we want all Kheshtaks of sequence 0 which is 0.


Duff is addicted to meat! Malek wants to keep her happy for n days. In order to be happy in i-th day, she needs to eat exactly ai kilograms of meat.  There is a big shop uptown and Malek wants to buy meat for her from there. In i-th day, they sell meat for pi dollars per kilogram. Malek knows all numbers a1, ..., an and p1, ..., pn. In each day, he can buy arbitrary amount of meat, also he can keep some meat he has for the future.Malek is a little tired from cooking meat, so he asked for your help. Help him to minimize the total money he spends to keep Duff happy for n days.

The first line of input contains integer n (1 ≤ n ≤ 105), the number of days. In the next n lines, i-th line contains two integers ai and pi (1 ≤ ai, pi ≤ 100), the amount of meat Duff needs and the cost of meat in that day.

Print the minimum money needed to keep Duff happy for n days, in one line.

NoteIn the first sample case: An optimal way would be to buy 1 kg on the first day, 2 kg on the second day and 3 kg on the third day.In the second sample case: An optimal way would be to buy 1 kg on the first day and 5 kg (needed meat for the second and third day) on the second day.


Duff is in love with lovely numbers! A positive integer x is called lovely if and only if there is no such positive integer a > 1 such that a2 is a divisor of x.  Malek has a number store! In his store, he has only divisors of positive integer n (and he has all of them). As a birthday present, Malek wants to give her a lovely number from his store. He wants this number to be as big as possible.Malek always had issues in math, so he asked for your help. Please tell him what is the biggest lovely number in his store.

The first and only line of input contains one integer, n (1 ≤ n ≤ 1012).

NoteIn first sample case, there are numbers 1, 2, 5 and 10 in the shop. 10 isn't divisible by any perfect square, so 10 is lovely.In second sample case, there are numbers 1, 2, 3, 4, 6 and 12 in the shop. 12 is divisible by 4 = 22, so 12 is not lovely, while 6 is indeed lovely.


A schoolboy named Vasya loves reading books on programming and mathematics. He has recently read an encyclopedia article that described the method of median smoothing (or median filter) and its many applications in science and engineering. Vasya liked the idea of the method very much, and he decided to try it in practice.Applying the simplest variant of median smoothing to the sequence of numbers a1, a2, ..., an will result a new sequence b1, b2, ..., bn obtained by the following algorithm:  b1 = a1, bn = an, that is, the first and the last number of the new sequence match the corresponding numbers of the original sequence.  For i = 2, ..., n - 1 value bi is equal to the median of three values ai - 1, ai and ai + 1. The median of a set of three numbers is the number that goes on the second place, when these three numbers are written in the non-decreasing order. For example, the median of the set 5, 1, 2 is number 2, and the median of set 1, 0, 1 is equal to 1.In order to make the task easier, Vasya decided to apply the method to sequences consisting of zeros and ones only.Having made the procedure once, Vasya looked at the resulting sequence and thought: what if I apply the algorithm to it once again, and then apply it to the next result, and so on? Vasya tried a couple of examples and found out that after some number of median smoothing algorithm applications the sequence can stop changing. We say that the sequence is stable, if it does not change when the median smoothing is applied to it.Now Vasya wonders, whether the sequence always eventually becomes stable. He asks you to write a program that, given a sequence of zeros and ones, will determine whether it ever becomes stable. Moreover, if it ever becomes stable, then you should determine what will it look like and how many times one needs to apply the median smoothing algorithm to initial sequence in order to obtain a stable one.

The first input line of the input contains a single integer n (3 ≤ n ≤ 500 000) — the length of the initial sequence. The next line contains n integers a1, a2, ..., an (ai = 0 or ai = 1), giving the initial sequence itself.

If the sequence will never become stable, print a single number  - 1. Otherwise, first print a single integer — the minimum number of times one needs to apply the median smoothing algorithm to the initial sequence before it becomes is stable. In the second line print n numbers separated by a space  — the resulting sequence itself.

NoteIn the second sample the stabilization occurs in two steps: , and the sequence 00000 is obviously stable.


A team of furry rescue rangers was sitting idle in their hollow tree when suddenly they received a signal of distress. In a few moments they were ready, and the dirigible of the rescue chipmunks hit the road.We assume that the action takes place on a Cartesian plane. The headquarters of the rescuers is located at point (x1, y1), and the distress signal came from the point (x2, y2).Due to Gadget's engineering talent, the rescuers' dirigible can instantly change its current velocity and direction of movement at any moment and as many times as needed. The only limitation is: the speed of the aircraft relative to the air can not exceed  meters per second.Of course, Gadget is a true rescuer and wants to reach the destination as soon as possible. The matter is complicated by the fact that the wind is blowing in the air and it affects the movement of the dirigible. According to the weather forecast, the wind will be defined by the vector (vx, vy) for the nearest t seconds, and then will change to (wx, wy). These vectors give both the direction and velocity of the wind. Formally, if a dirigible is located at the point (x, y), while its own velocity relative to the air is equal to zero and the wind (ux, uy) is blowing, then after  seconds the new position of the dirigible will be .Gadget is busy piloting the aircraft, so she asked Chip to calculate how long will it take them to reach the destination if they fly optimally. He coped with the task easily, but Dale is convinced that Chip has given the random value, aiming only not to lose the face in front of Gadget. Dale has asked you to find the right answer.It is guaranteed that the speed of the wind at any moment of time is strictly less than the maximum possible speed of the airship relative to the air.

The first line of the input contains four integers x1, y1, x2, y2 (|x1|,  |y1|,  |x2|,  |y2| ≤ 10 000) — the coordinates of the rescuers' headquarters and the point, where signal of the distress came from, respectively.  The second line contains two integers  and t (0 < v, t ≤ 1000), which are denoting the maximum speed of the chipmunk dirigible relative to the air and the moment of time when the wind changes according to the weather forecast, respectively.  Next follow one per line two pairs of integer (vx, vy) and (wx, wy), describing the wind for the first t seconds and the wind that will blow at all the remaining time, respectively. It is guaranteed that  and .

Print a single real value — the minimum time the rescuers need to get to point (x2, y2). You answer will be considered correct if its absolute or relative error does not exceed 10 - 6.  Namely: let's assume that your answer is a, and the answer of the jury is b. The checker program will consider your answer correct, if .


Duff is mad at her friends. That's why she sometimes makes Malek to take candy from one of her friends for no reason!  She has n friends. Her i-th friend's name is si (their names are not necessarily unique). q times, she asks Malek to take candy from her friends. She's angry, but also she acts with rules. When she wants to ask Malek to take candy from one of her friends, like k, she chooses two numbers l and r and tells Malek to take exactly  candies from him/her, where occur(t, s) is the number of occurrences of string t in s.Malek is not able to calculate how many candies to take in each request from Duff. That's why she asked for your help. Please tell him how many candies to take in each request.

The first line of input contains two integers n and q (1 ≤ n, q ≤ 105). The next n lines contain the names. i-th of them contains an string si, consisting of lowercase English letters   (). The next q lines contain the requests. Each of them contains three integers, l, r and k (says that Malek should take  candies from Duff's k-th friend).

Print the answer to each request in one line.


The famous global economic crisis is approaching rapidly, so the states of Berman, Berance and Bertaly formed an alliance and allowed the residents of all member states to freely pass through the territory of any of them. In addition, it was decided that a road between the states should be built to guarantee so that one could any point of any country can be reached from any point of any other State.Since roads are always expensive, the governments of the states of the newly formed alliance asked you to help them assess the costs. To do this, you have been issued a map that can be represented as a rectangle table consisting of n rows and m columns. Any cell of the map either belongs to one of three states, or is an area where it is allowed to build a road, or is an area where the construction of the road is not allowed. A cell is called passable, if it belongs to one of the states, or the road was built in this cell. From any passable cells you can move up, down, right and left, if the cell that corresponds to the movement exists and is passable.Your task is to construct a road inside a minimum number of cells, so that it would be possible to get from any cell of any state to any cell of any other state using only passable cells.It is guaranteed that initially it is possible to reach any cell of any state from any cell of this state, moving only along its cells. It is also guaranteed that for any state there is at least one cell that belongs to it.

The first line of the input contains the dimensions of the map n and m (1 ≤ n, m ≤ 1000) — the number of rows and columns respectively. Each of the next n lines contain m characters, describing the rows of the map. Digits from 1 to 3 represent the accessory to the corresponding state. The character '.' corresponds to the cell where it is allowed to build a road and the character '#' means no construction is allowed in this cell.

Print a single integer — the minimum number of cells you need to build a road inside in order to connect all the cells of all states. If such a goal is unachievable, print -1.


A top-secret military base under the command of Colonel Zuev is expecting an inspection from the Ministry of Defence. According to the charter, each top-secret military base must include a top-secret troop that should... well, we cannot tell you exactly what it should do, it is a top secret troop at the end. The problem is that Zuev's base is missing this top-secret troop for some reasons.The colonel decided to deal with the problem immediately and ordered to line up in a single line all n soldiers of the base entrusted to him. Zuev knows that the loquacity of the i-th soldier from the left is equal to qi. Zuev wants to form the top-secret troop using k leftmost soldiers in the line, thus he wants their total loquacity to be as small as possible (as the troop should remain top-secret). To achieve this, he is going to choose a pair of consecutive soldiers and swap them. He intends to do so no more than s times. Note that any soldier can be a participant of such swaps for any number of times. The problem turned out to be unusual, and colonel Zuev asked you to help.Determine, what is the minimum total loquacity of the first k soldiers in the line, that can be achieved by performing no more than s swaps of two consecutive soldiers.

The first line of the input contains three positive integers n, k, s (1 ≤ k ≤ n ≤ 150, 1 ≤ s ≤ 109) — the number of soldiers in the line, the size of the top-secret troop to be formed and the maximum possible number of swap operations of the consecutive pair of soldiers, respectively. The second line of the input contains n integer qi (1 ≤ qi ≤ 1 000 000) — the values of loquacity of soldiers in order they follow in line from left to right.

Print a single integer — the minimum possible total loquacity of the top-secret troop.

NoteIn the first sample Colonel has to swap second and third soldiers, he doesn't really need the remaining swap. The resulting soldiers order is: (2, 1, 4). Minimum possible summary loquacity of the secret troop is 3. In the second sample Colonel will perform swaps in the following order:  (10, 1, 6 — 2, 5)  (10, 1, 2, 6 — 5) The resulting soldiers order is (10, 1, 2, 5, 6). Minimum possible summary loquacity is equal to 18.


Today is birthday of a Little Dasha — she is now 8 years old! On this occasion, each of her n friends and relatives gave her a ribbon with a greeting written on it, and, as it turned out, all the greetings are different. Dasha gathered all the ribbons and decided to throw away some of them in order to make the remaining set stylish. The birthday girl considers a set of ribbons stylish if no greeting written on some ribbon is a substring of another greeting written on some other ribbon. Let us recall that the substring of the string s is a continuous segment of s.Help Dasha to keep as many ribbons as possible, so that she could brag about them to all of her friends. Dasha cannot rotate or flip ribbons, that is, each greeting can be read in a single way given in the input.

The first line of the input contains integer n (1 ≤ n ≤ 750) — the number of Dasha's relatives and friends. Each of the next n lines contains exactly one greeting. Each greeting consists of characters 'a' and 'b' only. The total length of all greetings won't exceed 10 000 000 characters.

In the first line print the maximum size of the stylish set. In the second line print the numbers of ribbons involved in it, assuming that they are numbered from 1 to n in the order they appear in the input. If there are several stylish sets of the maximum size, print any of them.

NoteIn the sample, the answer that keeps ribbons 3 and 4 is also considered correct.


Harry Potter and He-Who-Must-Not-Be-Named engaged in a fight to the death once again. This time they are located at opposite ends of the corridor of length l. Two opponents simultaneously charge a deadly spell in the enemy. We know that the impulse of Harry's magic spell flies at a speed of p meters per second, and the impulse of You-Know-Who's magic spell flies at a speed of q meters per second.The impulses are moving through the corridor toward each other, and at the time of the collision they turn round and fly back to those who cast them without changing their original speeds. Then, as soon as the impulse gets back to it's caster, the wizard reflects it and sends again towards the enemy, without changing the original speed of the impulse.Since Harry has perfectly mastered the basics of magic, he knows that after the second collision both impulses will disappear, and a powerful explosion will occur exactly in the place of their collision. However, the young wizard isn't good at math, so he asks you to calculate the distance from his position to the place of the second meeting of the spell impulses, provided that the opponents do not change positions during the whole fight.

The first line of the input contains a single integer l (1 ≤ l ≤ 1 000) — the length of the corridor where the fight takes place. The second line contains integer p, the third line contains integer q (1 ≤ p, q ≤ 500) — the speeds of magical impulses for Harry Potter and He-Who-Must-Not-Be-Named, respectively.

Print a single real number — the distance from the end of the corridor, where Harry is located, to the place of the second meeting of the spell impulses. Your answer will be considered correct if its absolute or relative error will not exceed 10 - 4.  Namely: let's assume that your answer equals a, and the answer of the jury is b. The checker program will consider your answer correct if .

NoteIn the first sample the speeds of the impulses are equal, so both of their meetings occur exactly in the middle of the corridor.


The name of one small but proud corporation consists of n lowercase English letters. The Corporation has decided to try rebranding — an active marketing strategy, that includes a set of measures to change either the brand (both for the company and the goods it produces) or its components: the name, the logo, the slogan. They decided to start with the name.For this purpose the corporation has consecutively hired m designers. Once a company hires the i-th designer, he immediately contributes to the creation of a new corporation name as follows: he takes the newest version of the name and replaces all the letters xi by yi, and all the letters yi by xi. This results in the new version. It is possible that some of these letters do no occur in the string. It may also happen that xi coincides with yi. The version of the name received after the work of the last designer becomes the new name of the corporation.Manager Arkady has recently got a job in this company, but is already soaked in the spirit of teamwork and is very worried about the success of the rebranding. Naturally, he can't wait to find out what is the new name the Corporation will receive.Satisfy Arkady's curiosity and tell him the final version of the name.

The first line of the input contains two integers n and m (1 ≤ n, m ≤ 200 000) — the length of the initial name and the number of designers hired, respectively. The second line consists of n lowercase English letters and represents the original name of the corporation. Next m lines contain the descriptions of the designers' actions: the i-th of them contains two space-separated lowercase English letters xi and yi.

NoteIn the second sample the name of the corporation consecutively changes as follows:

11 6
abacabadaba
a b
b c
a d
e g
f a
b b


Ari the monster always wakes up very early with the first ray of the sun and the first thing she does is feeding her squirrel.Ari draws a regular convex polygon on the floor and numbers it's vertices 1, 2, ..., n in clockwise order. Then starting from the vertex 1 she draws a ray in the direction of each other vertex. The ray stops when it reaches a vertex or intersects with another ray drawn before. Ari repeats this process for vertex 2, 3, ..., n (in this particular order). And then she puts a walnut in each region inside the polygon.  Ada the squirrel wants to collect all the walnuts, but she is not allowed to step on the lines drawn by Ari. That means Ada have to perform a small jump if she wants to go from one region to another. Ada can jump from one region P to another region Q if and only if P and Q share a side or a corner.Assuming that Ada starts from outside of the picture, what is the minimum number of jumps she has to perform in order to collect all the walnuts?

The first and only line of the input contains a single integer n (3 ≤ n ≤ 54321) - the number of vertices of the regular polygon drawn by Ari.

Print the minimum number of jumps Ada should make to collect all the walnuts. Note, that she doesn't need to leave the polygon after.

NoteOne of the possible solutions for the first sample is shown on the picture above.


Vector Willman and Array Bolt are the two most famous athletes of Byteforces. They are going to compete in a race with a distance of L meters today.  Willman and Bolt have exactly the same speed, so when they compete the result is always a tie. That is a problem for the organizers because they want a winner. While watching previous races the organizers have noticed that Willman can perform only steps of length equal to w meters, and Bolt can perform only steps of length equal to b meters. Organizers decided to slightly change the rules of the race. Now, at the end of the racetrack there will be an abyss, and the winner will be declared the athlete, who manages to run farther from the starting point of the the racetrack (which is not the subject to change by any of the athletes). Note that none of the athletes can run infinitely far, as they both will at some moment of time face the point, such that only one step further will cause them to fall in the abyss. In other words, the athlete will not fall into the abyss if the total length of all his steps will be less or equal to the chosen distance L.Since the organizers are very fair, the are going to set the length of the racetrack as an integer chosen randomly and uniformly in range from 1 to t (both are included). What is the probability that Willman and Bolt tie again today?

The first line of the input contains three integers t, w and b (1 ≤ t, w, b ≤ 5·1018) — the maximum possible length of the racetrack, the length of Willman's steps and the length of Bolt's steps respectively.

Print the answer to the problem as an irreducible fraction . Follow the format of the samples output. The fraction  (p and q are integers, and both p ≥ 0 and q > 0 holds) is called irreducible, if there is no such integer d > 1, that both p and q are divisible by d.

NoteIn the first sample Willman and Bolt will tie in case 1, 6 or 7 are chosen as the length of the racetrack.


Ari the monster is not an ordinary monster. She is the hidden identity of Super M, the Byteforces’ superhero. Byteforces is a country that consists of n cities, connected by n - 1 bidirectional roads. Every road connects exactly two distinct cities, and the whole road system is designed in a way that one is able to go from any city to any other city using only the given roads. There are m cities being attacked by humans. So Ari... we meant Super M have to immediately go to each of the cities being attacked to scare those bad humans. Super M can pass from one city to another only using the given roads. Moreover, passing through one road takes her exactly one kron - the time unit used in Byteforces.   However, Super M is not on Byteforces now - she is attending a training camp located in a nearby country Codeforces. Fortunately, there is a special device in Codeforces that allows her to instantly teleport from Codeforces to any city of Byteforces. The way back is too long, so for the purpose of this problem teleportation is used exactly once.You are to help Super M, by calculating the city in which she should teleport at the beginning in order to end her job in the minimum time (measured in krons). Also, provide her with this time so she can plan her way back to Codeforces.

The first line of the input contains two integers n and m (1 ≤ m ≤ n ≤ 123456) - the number of cities in Byteforces, and the number of cities being attacked respectively. Then follow n - 1 lines, describing the road system. Each line contains two city numbers ui and vi (1 ≤ ui, vi ≤ n) - the ends of the road i. The last line contains m distinct integers - numbers of cities being attacked. These numbers are given in no particular order.

First print the number of the city Super M should teleport to. If there are many possible optimal answers, print the one with the lowest city number. Then print the minimum possible time needed to scare all humans in cities being attacked, measured in Krons. Note that the correct answer is always unique.

NoteIn the first sample, there are two possibilities to finish the Super M's job in 3 krons. They are: and .However, you should choose the first one as it starts in the city with the lower number.


BCPC stands for Byteforces Collegiate Programming Contest, and is the most famous competition in Byteforces.BCPC is a team competition. Each team is composed by a coach and three contestants. Blenda is the coach of the Bit State University(BSU), and she is very strict selecting the members of her team.  In BSU there are n students numbered from 1 to n. Since all BSU students are infinitely smart, the only important parameters for Blenda are their reading and writing speed. After a careful measuring, Blenda have found that the i-th student have a reading speed equal to ri (words per minute), and a writing speed of wi (symbols per minute). Since BSU students are very smart, the measured speeds are sometimes very big and Blenda have decided to subtract some constant value c from all the values of reading speed and some value d from all the values of writing speed. Therefore she considers ri' = ri - c and wi' = wi - d. The student i is said to overwhelm the student j if and only if ri'·wj' > rj'·wi'. Blenda doesn’t like fights in teams, so she thinks that a team consisting of three distinct students i, j and k is good if i overwhelms j, j overwhelms k, and k overwhelms i. Yes, the relation of overwhelming is not transitive as it often happens in real life.Since Blenda is busy preparing a training camp in Codeforces, you are given a task to calculate the number of different good teams in BSU. Two teams are considered to be different if there is at least one student that is present in one team but is not present in the other. In other words, two teams are different if the sets of students that form these teams are different.

In the first line of the input three integers n, c and d (3 ≤ n ≤ 345678, 1 ≤ c, d ≤ 109) are written. They denote the number of students Blenda can use to form teams, the value subtracted from all reading speeds and the value subtracted from all writing speeds respectively. Each of the next n lines contains two integers ri and wi (0 < ri, wi ≤ 109, |ri - c| + |wi - d| > 0). There are no two students, such that both their reading and writing speeds coincide, i.e. for every i ≠ j condition |ri - rj| + |wi - wj| > 0 holds.

Print the number of different teams in BSU, that are good according to Blenda's definition.

NoteIn the first sample the following teams are good: (i = 1, j = 2, k = 3), (i = 2, j = 5, k = 1), (i = 1, j = 4, k = 3), (i = 5, j = 1, k = 4).Note, that for example the team (i = 3, j = 1, k = 2) is also good, but is considered to be the same as the team (i = 1, j = 2, k = 3).


Galois is one of the strongest chess players of Byteforces. He has even invented a new variant of chess, which he named «PawnChess».This new game is played on a board consisting of 8 rows and 8 columns. At the beginning of every game some black and white pawns are placed on the board. The number of black pawns placed is not necessarily equal to the number of white pawns placed.   Lets enumerate rows and columns with integers from 1 to 8. Rows are numbered from top to bottom, while columns are numbered from left to right. Now we denote as (r, c) the cell located at the row r and at the column c.There are always two players A and B playing the game. Player A plays with white pawns, while player B plays with black ones. The goal of player A is to put any of his pawns to the row 1, while player B tries to put any of his pawns to the row 8. As soon as any of the players completes his goal the game finishes immediately and the succeeded player is declared a winner.Player A moves first and then they alternate turns. On his move player A must choose exactly one white pawn and move it one step upward and player B (at his turn) must choose exactly one black pawn and move it one step down. Any move is possible only if the targeted cell is empty. It's guaranteed that for any scenario of the game there will always be at least one move available for any of the players.Moving upward means that the pawn located in (r, c) will go to the cell (r - 1, c), while moving down means the pawn located in (r, c) will go to the cell (r + 1, c). Again, the corresponding cell must be empty, i.e. not occupied by any other pawn of any color.Given the initial disposition of the board, determine who wins the game if both players play optimally. Note that there will always be a winner due to the restriction that for any game scenario both players will have some moves available.

The input consists of the board description given in eight lines, each line contains eight characters. Character 'B' is used to denote a black pawn, and character 'W' represents a white pawn. Empty cell is marked with '.'.  It's guaranteed that there will not be white pawns on the first row neither black pawns on the last row.

Print 'A' if player A wins the game on the given board, and 'B' if player B will claim the victory. Again, it's guaranteed that there will always be a winner on the given board.

NoteIn the first sample player A is able to complete his goal in 3 steps by always moving a pawn initially located at (4, 5). Player B needs at least 5 steps for any of his pawns to reach the row 8. Hence, player A will be the winner.


Andrew often reads articles in his favorite magazine 2Char. The main feature of these articles is that each of them uses at most two distinct letters. Andrew decided to send an article to the magazine, but as he hasn't written any article, he just decided to take a random one from magazine 26Char. However, before sending it to the magazine 2Char, he needs to adapt the text to the format of the journal. To do so, he removes some words from the chosen article, in such a way that the remaining text can be written using no more than two distinct letters.Since the payment depends from the number of non-space characters in the article, Andrew wants to keep the words with the maximum total length.

The first line of the input contains number n (1 ≤ n ≤ 100) — the number of words in the article chosen by Andrew. Following are n lines, each of them contains one word. All the words consist only of small English letters and their total length doesn't exceed 1000. The words are not guaranteed to be distinct, in this case you are allowed to use a word in the article as many times as it appears in the input.

Print a single integer — the maximum possible total length of words in Andrew's article.

NoteIn the first sample the optimal way to choose words is {'abb', 'aaa', 'bbb'}.In the second sample the word 'cdecdecdecdecdecde' consists of three distinct letters, and thus cannot be used in the article. The optimal answer is {'a', 'a', 'aaaa'}.


The teacher gave Anton a large geometry homework, but he didn't do it (as usual) as he participated in a regular round on Codeforces. In the task he was given a set of n lines defined by the equations y = ki·x + bi. It was necessary to determine whether there is at least one point of intersection of two of these lines, that lays strictly inside the strip between x1 < x2. In other words, is it true that there are 1 ≤ i < j ≤ n and x', y', such that:   y' = ki * x' + bi, that is, point (x', y') belongs to the line number i;  y' = kj * x' + bj, that is, point (x', y') belongs to the line number j;  x1 < x' < x2, that is, point (x', y') lies inside the strip bounded by x1 < x2. You can't leave Anton in trouble, can you? Write a program that solves the given task.

The first line of the input contains an integer n (2 ≤ n ≤ 100 000) — the number of lines in the task given to Anton. The second line contains integers x1 and x2 ( - 1 000 000 ≤ x1 < x2 ≤ 1 000 000) defining the strip inside which you need to find a point of intersection of at least two lines. The following n lines contain integers ki, bi ( - 1 000 000 ≤ ki, bi ≤ 1 000 000) — the descriptions of the lines. It is guaranteed that all lines are pairwise distinct, that is, for any two i ≠ j it is true that either ki ≠ kj, or bi ≠ bj.

Print "Yes" (without quotes), if there is at least one intersection of two distinct lines, located strictly inside the strip. Otherwise print "No" (without quotes).

NoteIn the first sample there are intersections located on the border of the strip, but there are no intersections located strictly inside it.


Every day Ruslan tried to count sheep to fall asleep, but this didn't help. Now he has found a more interesting thing to do. First, he thinks of some set of circles on a plane, and then tries to choose a beautiful set of points, such that there is at least one point from the set inside or on the border of each of the imagined circles.Yesterday Ruslan tried to solve this problem for the case when the set of points is considered beautiful if it is given as (xt = f(t), yt = g(t)), where argument t takes all integer values from 0 to 50. Moreover, f(t) and g(t) should be correct functions.Assume that w(t) and h(t) are some correct functions, and c is an integer ranging from 0 to 50. The function s(t) is correct if it's obtained by one of the following rules:  s(t) = abs(w(t)), where abs(x) means taking the absolute value of a number x, i.e. |x|; s(t) = (w(t) + h(t)); s(t) = (w(t) - h(t)); s(t) = (w(t) * h(t)), where  *  means multiplication, i.e. (w(t)·h(t)); s(t) = c; s(t) = t;Yesterday Ruslan thought on and on, but he could not cope with the task. Now he asks you to write a program that computes the appropriate f(t) and g(t) for any set of at most 50 circles.In each of the functions f(t) and g(t) you are allowed to use no more than 50 multiplications. The length of any function should not exceed 100·n characters. The function should not contain spaces.Ruslan can't keep big numbers in his memory, so you should choose f(t) and g(t), such that for all integer t from 0 to 50 value of f(t) and g(t) and all the intermediate calculations won't exceed 109 by their absolute value.

The first line of the input contains number n (1 ≤ n ≤ 50) — the number of circles Ruslan thinks of. Next follow n lines, each of them containing three integers xi, yi and ri (0 ≤ xi, yi ≤ 50, 2 ≤ ri ≤ 50) — the coordinates of the center and the raduis of the i-th circle.

In the first line print a correct function f(t). In the second line print a correct function g(t). The set of the points (xt = f(t), yt = g(t)) (0 ≤ t ≤ 50) must satisfy the condition, that there is at least one point inside or on the border of each of the circles, Ruslan thinks of at the beginning.

NoteCorrect functions: 10 (1+2) ((t-3)+(t*4)) abs((t-10)) (abs((((23-t)*(t*t))+((45+12)*(t*t))))*((5*t)+((12*t)-13))) abs((t-(abs((t*31))+14))))Incorrect functions: 3+5+7 (not enough brackets, it should be ((3+5)+7) or (3+(5+7)))  abs(t-3) (not enough brackets, it should be abs((t-3)) 2+(2-3 (one bracket too many) 1(t+5) (no arithmetic operation between 1 and the bracket) 5000*5000 (the number exceeds the maximum)  The picture shows one of the possible solutions


Bogdan has a birthday today and mom gave him a tree consisting of n vertecies. For every edge of the tree i, some number xi was written on it. In case you forget, a tree is a connected non-directed graph without cycles. After the present was granted, m guests consecutively come to Bogdan's party. When the i-th guest comes, he performs exactly one of the two possible operations:   Chooses some number yi, and two vertecies ai and bi. After that, he moves along the edges of the tree from vertex ai to vertex bi using the shortest path (of course, such a path is unique in the tree). Every time he moves along some edge j, he replaces his current number yi by , that is, by the result of integer division yi div xj.  Chooses some edge pi and replaces the value written in it xpi by some positive integer ci < xpi. As Bogdan cares about his guests, he decided to ease the process. Write a program that performs all the operations requested by guests and outputs the resulting value yi for each i of the first type.

The first line of the input contains integers, n and m (2 ≤ n ≤ 200 000, 1 ≤ m ≤ 200 000) — the number of vertecies in the tree granted to Bogdan by his mom and the number of guests that came to the party respectively. Next n - 1 lines contain the description of the edges. The i-th of these lines contains three integers ui, vi and xi (1 ≤ ui, vi ≤ n, ui ≠ vi, 1 ≤ xi ≤ 1018), denoting an edge that connects vertecies ui and vi, with the number xi initially written on it. The following m lines describe operations, requested by Bogdan's guests. Each description contains three or four integers and has one of the two possible forms:    1 ai bi yi corresponds to a guest, who chooses the operation of the first type.  2 pi ci corresponds to a guests, who chooses the operation of the second type.  1 ≤ ai, bi ≤ n 1 ≤ pi ≤ n - 1 1 ≤ yi ≤ 1018 1 ≤ ci < xpi xpi pi xpi 1 n - 1

For each guest who chooses the operation of the first type, print the result of processing the value yi through the path from ai to bi.

NoteInitially the tree looks like this:   The response to the first query is:  = 2After the third edge is changed, the tree looks like this:   The response to the second query is:  = 4In the third query the initial and final vertex coincide, that is, the answer will be the initial number 20.After the change in the fourth edge the tree looks like this:   In the last query the answer will be:  = 3


Gosha's universe is a table consisting of n rows and m columns. Both the rows and columns are numbered with consecutive integers starting with 1. We will use (r, c) to denote a cell located in the row r and column c.Gosha is often invited somewhere. Every time he gets an invitation, he first calculates the number of ways to get to this place, and only then he goes. Gosha's house is located in the cell (1, 1).At any moment of time, Gosha moves from the cell he is currently located in to a cell adjacent to it (two cells are adjacent if they share a common side). Of course, the movement is possible only if such a cell exists, i.e. Gosha will not go beyond the boundaries of the table. Thus, from the cell (r, c) he is able to make a move to one of the cells (r - 1, c), (r, c - 1), (r + 1, c), (r, c + 1). Also, Ghosha can skip a move and stay in the current cell (r, c).Besides the love of strange calculations, Gosha is allergic to cats, so he never goes to the cell that has a cat in it. Gosha knows exactly where and when he will be invited and the schedule of cats travelling along the table. Formally, he has q records, the i-th of them has one of the following forms:   1, xi, yi, ti — Gosha is invited to come to cell (xi, yi) at the moment of time ti. It is guaranteed that there is no cat inside cell (xi, yi) at this moment of time.  2, xi, yi, ti — at the moment ti a cat appears in cell (xi, yi). It is guaranteed that no other cat is located in this cell (xi, yi) at that moment of time.  3, xi, yi, ti — at the moment ti a cat leaves cell (xi, yi). It is guaranteed that there is cat located in the cell (xi, yi). Gosha plans to accept only one invitation, but he has not yet decided, which particular one. In order to make this decision, he asks you to calculate for each of the invitations i the number of ways to get to the cell (xi, yi) at the moment ti. For every invitation, assume that Gosha he starts moving from cell (1, 1) at the moment 1.Moving between two neighboring cells takes Gosha exactly one unit of tim. In particular, this means that Gosha can come into the cell only if a cat sitting in it leaves the moment when Gosha begins his movement from the neighboring cell, and if none of the cats comes to the cell at the time when Gosha is in it.Two ways to go from cell (1, 1) to cell (x, y) at time t are considered distinct if for at least one moment of time from 1 to t Gosha's positions are distinct for the two ways at this moment. Note, that during this travel Gosha is allowed to visit both (1, 1) and (x, y) multiple times. Since the number of ways can be quite large, print it modulo 109 + 7.

The first line of the input contains three positive integers n, m and q (1 ≤ n·m ≤ 20, 1 ≤ q ≤ 10 000) — the number of rows and columns in the table and the number of events respectively. Next q lines describe the events, each description contains four integers tpi, xi, yi and ti (1 ≤ tp ≤ 3, 1 ≤ x ≤ n, 1 ≤ y ≤ m, 2 ≤ t ≤ 109) — the type of the event (1 if Gosha gets an invitation, 2 if a cat comes to the cell and 3 if a cat leaves the cell), the coordinates of the cell where the action takes place and the moment of time at which the action takes place respectively. It is guaranteed that the queries are given in the chronological order, i.e. ti < ti + 1.

For each invitation i (that is, tpi = 1) calculate the number of ways to get to cell (xi, yi) at the moment of time ti. Respond to the invitations chronologically, that is, in the order they appear in the input.

NoteExplanation of the first sample. Each picture specifies the number of ways to arrive at the cell at the appropriate time. (X stands for a cell blocked at this particular moment of time)  Time moment 1.   Time moment 2.  Time moment 3.  Time moment 4.  Time moment 5.  Time moment 6.  Time moment 7.


For months Maxim has been coming to work on his favorite bicycle. And quite recently he decided that he is ready to take part in a cyclists' competitions.He knows that this year n competitions will take place. During the i-th competition the participant must as quickly as possible complete a ride along a straight line from point si to point fi (si < fi).Measuring time is a complex process related to usage of a special sensor and a time counter. Think of the front wheel of a bicycle as a circle of radius r. Let's neglect the thickness of a tire, the size of the sensor, and all physical effects. The sensor is placed on the rim of the wheel, that is, on some fixed point on a circle of radius r. After that the counter moves just like the chosen point of the circle, i.e. moves forward and rotates around the center of the circle.At the beginning each participant can choose any point bi, such that his bike is fully behind the starting line, that is, bi < si - r. After that, he starts the movement, instantly accelerates to his maximum speed and at time tsi, when the coordinate of the sensor is equal to the coordinate of the start, the time counter starts. The cyclist makes a complete ride, moving with his maximum speed and at the moment the sensor's coordinate is equal to the coordinate of the finish (moment of time tfi), the time counter deactivates and records the final time. Thus, the counter records that the participant made a complete ride in time tfi - tsi.  Maxim is good at math and he suspects that the total result doesn't only depend on his maximum speed v, but also on his choice of the initial point bi. Now Maxim is asking you to calculate for each of n competitions the minimum possible time that can be measured by the time counter. The radius of the wheel of his bike is equal to r.

The first line contains three integers n, r and v (1 ≤ n ≤ 100 000, 1 ≤ r, v ≤ 109) — the number of competitions, the radius of the front wheel of Max's bike and his maximum speed, respectively.  Next n lines contain the descriptions of the contests. The i-th line contains two integers si and fi (1 ≤ si < fi ≤ 109) — the coordinate of the start and the coordinate of the finish on the i-th competition.

Print n real numbers, the i-th number should be equal to the minimum possible time measured by the time counter. Your answer will be considered correct if its absolute or relative error will not exceed 10 - 6.  Namely: let's assume that your answer equals a, and the answer of the jury is b. The checker program will consider your answer correct if .


Edo has got a collection of n refrigerator magnets!He decided to buy a refrigerator and hang the magnets on the door. The shop can make the refrigerator with any size of the door that meets the following restrictions: the refrigerator door must be rectangle, and both the length and the width of the door must be positive integers.Edo figured out how he wants to place the magnets on the refrigerator. He introduced a system of coordinates on the plane, where each magnet is represented as a rectangle with sides parallel to the coordinate axes.Now he wants to remove no more than k magnets (he may choose to keep all of them) and attach all remaining magnets to the refrigerator door, and the area of the door should be as small as possible. A magnet is considered to be attached to the refrigerator door if its center lies on the door or on its boundary. The relative positions of all the remaining magnets must correspond to the plan.Let us explain the last two sentences. Let's suppose we want to hang two magnets on the refrigerator. If the magnet in the plan has coordinates of the lower left corner (x1, y1) and the upper right corner (x2, y2), then its center is located at (, ) (may not be integers). By saying the relative position should correspond to the plan we mean that the only available operation is translation, i.e. the vector connecting the centers of two magnets in the original plan, must be equal to the vector connecting the centers of these two magnets on the refrigerator.The sides of the refrigerator door must also be parallel to coordinate axes.

The first line contains two integers n and k (1 ≤ n ≤ 100 000, 0 ≤ k ≤ min(10, n - 1)) — the number of magnets that Edo has and the maximum number of magnets Edo may not place on the refrigerator. Next n lines describe the initial plan of placing magnets. Each line contains four integers x1, y1, x2, y2 (1 ≤ x1 < x2 ≤ 109, 1 ≤ y1 < y2 ≤ 109) — the coordinates of the lower left and upper right corners of the current magnet. The magnets can partially overlap or even fully coincide.

Print a single integer — the minimum area of the door of refrigerator, which can be used to place at least n - k magnets, preserving the relative positions.

NoteIn the first test sample it is optimal to remove either the first or the third magnet. If we remove the first magnet, the centers of two others will lie at points (2.5, 2.5) and (3.5, 3.5). Thus, it is enough to buy a fridge with door width 1 and door height 1, the area of the door also equals one, correspondingly.In the second test sample it doesn't matter which magnet to remove, the answer will not change — we need a fridge with door width 8 and door height 8.In the third sample you cannot remove anything as k = 0.


Today on a math lesson the teacher told Vovochka that the Euler function of a positive integer φ(n) is an arithmetic function that counts the positive integers less than or equal to n that are relatively prime to n. The number 1 is coprime to all the positive integers and φ(1) = 1.Now the teacher gave Vovochka an array of n positive integers a1, a2, ..., an and a task to process q queries li ri — to calculate and print  modulo 109 + 7. As it is too hard for a second grade school student, you've decided to help Vovochka.

The first line of the input contains number n (1 ≤ n ≤ 200 000) — the length of the array given to Vovochka. The second line contains n integers a1, a2, ..., an (1 ≤ ai ≤ 106). The third line contains integer q (1 ≤ q ≤ 200 000) — the number of queries. Next q lines contain the queries, one per line. Each query is defined by the boundaries of the segment li and ri (1 ≤ li ≤ ri ≤ n).

Print q numbers — the value of the Euler function for each query, calculated modulo 109 + 7.


You are given a non-empty line s and an integer k. The following operation is performed with this line exactly once:  A line is split into at most k non-empty substrings, i.e. string s is represented as a concatenation of a set of strings s = t1 + t2 + ... + tm, 1 ≤ m ≤ k.  Some of strings ti are replaced by strings tir, that is, their record from right to left.  The lines are concatenated back in the same order, we get string s' = t'1t'2... t'm, where t'i equals ti or tir. Your task is to determine the lexicographically smallest string that could be the result of applying the given operation to the string s.

The first line of the input contains string s (1 ≤ |s| ≤ 5 000 000), consisting of lowercase English letters. The second line contains integer k (1 ≤ k ≤ |s|) — the maximum number of parts in the partition.

In the single line print the lexicographically minimum string s' which can be obtained as a result of performing the described operation.


One day Vitaly was going home late at night and wondering: how many people aren't sleeping at that moment? To estimate, Vitaly decided to look which windows are lit in the house he was passing by at that moment.Vitaly sees a building of n floors and 2·m windows on each floor. On each floor there are m flats numbered from 1 to m, and two consecutive windows correspond to each flat. If we number the windows from 1 to 2·m from left to right, then the j-th flat of the i-th floor has windows 2·j - 1 and 2·j in the corresponding row of windows (as usual, floors are enumerated from the bottom). Vitaly thinks that people in the flat aren't sleeping at that moment if at least one of the windows corresponding to this flat has lights on.Given the information about the windows of the given house, your task is to calculate the number of flats where, according to Vitaly, people aren't sleeping.

The first line of the input contains two integers n and m (1 ≤ n, m ≤ 100) — the number of floors in the house and the number of flats on each floor respectively. Next n lines describe the floors from top to bottom and contain 2·m characters each. If the i-th window of the given floor has lights on, then the i-th character of this line is '1', otherwise it is '0'.

Print a single integer — the number of flats that have lights on in at least one window, that is, the flats where, according to Vitaly, people aren't sleeping.

NoteIn the first test case the house has two floors, two flats on each floor. That is, in total there are 4 flats. The light isn't on only on the second floor in the left flat. That is, in both rooms of the flat the light is off.In the second test case the house has one floor and the first floor has three flats. The light is on in the leftmost flat (in both windows) and in the middle flat (in one window). In the right flat the light is off.


In the official contest this problem has a different statement, for which jury's solution was working incorrectly, and for this reason it was excluded from the contest. This mistake have been fixed and the current given problem statement and model solution corresponds to what jury wanted it to be during the contest.Vova and Lesha are friends. They often meet at Vova's place and compete against each other in a computer game named The Ancient Papyri: Swordsink. Vova always chooses a warrior as his fighter and Leshac chooses an archer. After that they should choose initial positions for their characters and start the fight. A warrior is good at melee combat, so Vova will try to make the distance between fighters as small as possible. An archer prefers to keep the enemy at a distance, so Lesha will try to make the initial distance as large as possible.There are n (n is always even) possible starting positions for characters marked along the Ox axis. The positions are given by their distinct coordinates x1, x2, ..., xn, two characters cannot end up at the same position.Vova and Lesha take turns banning available positions, Vova moves first. During each turn one of the guys bans exactly one of the remaining positions. Banned positions cannot be used by both Vova and Lesha. They continue to make moves until there are only two possible positions remaining (thus, the total number of moves will be n - 2). After that Vova's character takes the position with the lesser coordinate and Lesha's character takes the position with the bigger coordinate and the guys start fighting.Vova and Lesha are already tired by the game of choosing positions, as they need to play it before every fight, so they asked you (the developer of the The Ancient Papyri: Swordsink) to write a module that would automatically determine the distance at which the warrior and the archer will start fighting if both Vova and Lesha play optimally.

The first line on the input contains a single integer n (2 ≤ n ≤ 200 000, n is even) — the number of positions available initially. The second line contains n distinct integers x1, x2, ..., xn (0 ≤ xi ≤ 109), giving the coordinates of the corresponding positions.

Print the distance between the warrior and the archer at the beginning of the fight, provided that both Vova and Lesha play optimally.

NoteIn the first sample one of the optimum behavior of the players looks like that:  Vova bans the position at coordinate 15;  Lesha bans the position at coordinate 3;  Vova bans the position at coordinate 31;  Lesha bans the position at coordinate 1. After these actions only positions 0 and 7 will remain, and the distance between them is equal to 7.In the second sample there are only two possible positions, so there will be no bans.


Pasha has recently bought a new phone jPager and started adding his friends' phone numbers there. Each phone number consists of exactly n digits.Also Pasha has a number k and two sequences of length n / k (n is divisible by k) a1, a2, ..., an / k and b1, b2, ..., bn / k. Let's split the phone number into blocks of length k. The first block will be formed by digits from the phone number that are on positions 1, 2,..., k, the second block will be formed by digits from the phone number that are on positions k + 1, k + 2, ..., 2·k and so on. Pasha considers a phone number good, if the i-th block doesn't start from the digit bi and is divisible by ai if represented as an integer. To represent the block of length k as an integer, let's write it out as a sequence c1, c2,...,ck. Then the integer is calculated as the result of the expression c1·10k - 1 + c2·10k - 2 + ... + ck.Pasha asks you to calculate the number of good phone numbers of length n, for the given k, ai and bi. As this number can be too big, print it modulo 109 + 7.

The first line of the input contains two integers n and k (1 ≤ n ≤ 100 000, 1 ≤ k ≤ min(n, 9)) — the length of all phone numbers and the length of each block, respectively. It is guaranteed that n is divisible by k. The second line of the input contains n / k space-separated positive integers — sequence a1, a2, ..., an / k (1 ≤ ai < 10k). The third line of the input contains n / k space-separated positive integers — sequence b1, b2, ..., bn / k (0 ≤ bi ≤ 9).

Print a single integer — the number of good phone numbers of length n modulo 109 + 7.


After making bad dives into swimming pools, Wilbur wants to build a swimming pool in the shape of a rectangle in his backyard. He has set up coordinate axes, and he wants the sides of the rectangle to be parallel to them. Of course, the area of the rectangle must be positive. Wilbur had all four vertices of the planned pool written on a paper, until his friend came along and erased some of the vertices.Now Wilbur is wondering, if the remaining n vertices of the initial rectangle give enough information to restore the area of the planned swimming pool.

The first line of the input contains a single integer n (1 ≤ n ≤ 4) — the number of vertices that were not erased by Wilbur's friend. Each of the following n lines contains two integers xi and yi ( - 1000 ≤ xi, yi ≤ 1000) —the coordinates of the i-th vertex that remains. Vertices are given in an arbitrary order. It's guaranteed that these points are distinct vertices of some rectangle, that has positive area and which sides are parallel to the coordinate axes.

Print the area of the initial rectangle if it could be uniquely determined by the points remaining. Otherwise, print  - 1.

NoteIn the first sample, two opposite corners of the initial rectangle are given, and that gives enough information to say that the rectangle is actually a unit square.In the second sample there is only one vertex left and this is definitely not enough to uniquely define the area.


Wilbur the pig is tinkering with arrays again. He has the array a1, a2, ..., an initially consisting of n zeros. At one step, he can choose any index i and either add 1 to all elements ai, ai + 1, ... , an or subtract 1 from all elements ai, ai + 1, ..., an. His goal is to end up with the array b1, b2, ..., bn. Of course, Wilbur wants to achieve this goal in the minimum number of steps and asks you to compute this value.

The first line of the input contains a single integer n (1 ≤ n ≤ 200 000) — the length of the array ai. Initially ai = 0 for every position i, so this array is not given in the input. The second line of the input contains n integers b1, b2, ..., bn ( - 109 ≤ bi ≤ 109).

Print the minimum number of steps that Wilbur needs to make in order to achieve ai = bi for all i.

NoteIn the first sample, Wilbur may successively choose indices 1, 2, 3, 4, and 5, and add 1 to corresponding suffixes.In the second sample, Wilbur first chooses indices 1 and 2 and adds 1 to corresponding suffixes, then he chooses index 4 and subtract 1.


Wilbur is playing with a set of n points on the coordinate plane. All points have non-negative integer coordinates. Moreover, if some point (x, y) belongs to the set, then all points (x', y'), such that 0 ≤ x' ≤ x and 0 ≤ y' ≤ y also belong to this set.Now Wilbur wants to number the points in the set he has, that is assign them distinct integer numbers from 1 to n. In order to make the numbering aesthetically pleasing, Wilbur imposes the condition that if some point (x, y) gets number i, then all (x',y') from the set, such that x' ≥ x and y' ≥ y must be assigned a number not less than i. For example, for a set of four points (0, 0), (0, 1), (1, 0) and (1, 1), there are two aesthetically pleasing numberings. One is 1, 2, 3, 4 and another one is 1, 3, 2, 4.Wilbur's friend comes along and challenges Wilbur. For any point he defines it's special value as s(x, y) = y - x. Now he gives Wilbur some w1, w2,..., wn, and asks him to find an aesthetically pleasing numbering of the points in the set, such that the point that gets number i has it's special value equal to wi, that is s(xi, yi) = yi - xi = wi.Now Wilbur asks you to help him with this challenge.

The first line of the input consists of a single integer n (1 ≤ n ≤ 100 000) — the number of points in the set Wilbur is playing with. Next follow n lines with points descriptions. Each line contains two integers x and y (0 ≤ x, y ≤ 100 000), that give one point in Wilbur's set. It's guaranteed that all points are distinct. Also, it is guaranteed that if some point (x, y) is present in the input, then all points (x', y'), such that 0 ≤ x' ≤ x and 0 ≤ y' ≤ y, are also present in the input. The last line of the input contains n integers. The i-th of them is wi ( - 100 000 ≤ wi ≤ 100 000) — the required special value of the point that gets number i in any aesthetically pleasing numbering.

If there exists an aesthetically pleasant numbering of points in the set, such that s(xi, yi) = yi - xi = wi, then print "YES" on the first line of the output. Otherwise, print "NO". If a solution exists, proceed output with n lines. On the i-th of these lines print the point of the set that gets number i. If there are multiple solutions, print any of them.

NoteIn the first sample, point (2, 0) gets number 3, point (0, 0) gets number one, point (1, 0) gets number 2, point (1, 1) gets number 5 and point (0, 1) gets number 4. One can easily check that this numbering is aesthetically pleasing and yi - xi = wi.In the second sample, the special values of the points in the set are 0,  - 1, and  - 2 while the sequence that the friend gives to Wilbur is 0, 1, 2. Therefore, the answer does not exist.


Wilbur the pig really wants to be a beaver, so he decided today to pretend he is a beaver and bite at trees to cut them down.There are n trees located at various positions on a line. Tree i is located at position xi. All the given positions of the trees are distinct.The trees are equal, i.e. each tree has height h. Due to the wind, when a tree is cut down, it either falls left with probability p, or falls right with probability 1 - p. If a tree hits another tree while falling, that tree will fall in the same direction as the tree that hit it. A tree can hit another tree only if the distance between them is strictly less than h. For example, imagine there are 4 trees located at positions 1, 3, 5 and 8, while h = 3 and the tree at position 1 falls right. It hits the tree at position 3 and it starts to fall too. In it's turn it hits the tree at position 5 and it also starts to fall. The distance between 8 and 5 is exactly 3, so the tree at position 8 will not fall.As long as there are still trees standing, Wilbur will select either the leftmost standing tree with probability 0.5 or the rightmost standing tree with probability 0.5. Selected tree is then cut down. If there is only one tree remaining, Wilbur always selects it. As the ground is covered with grass, Wilbur wants to know the expected total length of the ground covered with fallen trees after he cuts them all down because he is concerned about his grass-eating cow friends. Please help Wilbur.

The first line of the input contains two integers, n (1 ≤ n ≤ 2000) and h (1 ≤ h ≤ 108) and a real number p (0 ≤ p ≤ 1), given with no more than six decimal places. The second line of the input contains n integers, x1, x2, ..., xn ( - 108 ≤ xi ≤ 108) in no particular order.

Print a single real number — the expected total length of the ground covered by trees when they have all fallen down. Your answer will be considered correct if its absolute or relative error does not exceed 10 - 6.  Namely: let's assume that your answer is a, and the answer of the jury is b. The checker program will consider your answer correct, if .

NoteConsider the first example, we have 2 trees with height 2.    There are 3 scenarios:  1. Both trees falls left. This can either happen with the right tree falling left first, which has  probability (also knocking down the left tree), or the left tree can fall left and then the right tree can fall left, which has  probability. Total probability is .  2. Both trees fall right. This is analogous to (1), so the probability of this happening is .  3. The left tree fall left and the right tree falls right. This is the only remaining scenario so it must have  probability.  Cases 1 and 2 lead to a total of 3 units of ground covered, while case 3 leads to a total of 4 units of ground covered. Thus, the expected value is .


Find the number of k-divisible numbers on the segment [a, b]. In other words you need to find the number of such integer values x that a ≤ x ≤ b and x is divisible by k.

The only line contains three space-separated integers k, a and b (1 ≤ k ≤ 1018; - 1018 ≤ a ≤ b ≤ 1018).


For the given sequence with n different elements find the number of increasing subsequences with k + 1 elements. It is guaranteed that the answer is not greater than 8·1018.

First line contain two integer values n and k (1 ≤ n ≤ 105, 0 ≤ k ≤ 10) — the length of sequence and the number of elements in increasing subsequences. Next n lines contains one integer ai (1 ≤ ai ≤ n) each — elements of sequence. All values ai are different.

Print one integer — the answer to the problem.


A restaurant received n orders for the rental. Each rental order reserve the restaurant for a continuous period of time, the i-th order is characterized by two time values — the start time li and the finish time ri (li ≤ ri).Restaurant management can accept and reject orders. What is the maximal number of orders the restaurant can accept?No two accepted orders can intersect, i.e. they can't share even a moment of time. If one order ends in the moment other starts, they can't be accepted both.

The first line contains integer number n (1 ≤ n ≤ 5·105) — number of orders. The following n lines contain integer values li and ri each (1 ≤ li ≤ ri ≤ 109).

Print the maximal number of orders that can be accepted.


Wilbur the pig now wants to play with strings. He has found an n by m table consisting only of the digits from 0 to 9 where the rows are numbered 1 to n and the columns are numbered 1 to m. Wilbur starts at some square and makes certain moves. If he is at square (x, y) and the digit d (0 ≤ d ≤ 9) is written at position (x, y), then he must move to the square (x + ad, y + bd), if that square lies within the table, and he stays in the square (x, y) otherwise. Before Wilbur makes a move, he can choose whether or not to write the digit written in this square on the white board. All digits written on the whiteboard form some string. Every time a new digit is written, it goes to the end of the current string.Wilbur has q strings that he is worried about. For each string si, Wilbur wants to know whether there exists a starting position (x, y) so that by making finitely many moves, Wilbur can end up with the string si written on the white board.

The first line of the input consists of three integers n, m, and q (1 ≤ n, m, q ≤ 200) — the dimensions of the table and the number of strings to process, respectively. Each of the next n lines contains m digits from 0 and 9 giving the table itself. Then follow 10 lines. The i-th of them contains the values ai - 1 and bi - 1 ( - 200 ≤ ai, bi ≤ 200), i.e. the vector that Wilbur uses to make a move from the square with a digit i - 1 in it. There are q lines that follow. The i-th of them will contain a string si consisting only of digits from 0 to 9. It is guaranteed that the total length of these q strings won't exceed 1 000 000.

For each of the q strings, print "YES" if Wilbur can choose x and y in order to finish with this string after some finite number of moves. If it's impossible, than print "NO" for the corresponding string.

NoteIn the first sample, there is a 1 by 1 table consisting of the only digit 0. The only move that can be made is staying on the square. The first string can be written on the white board by writing 0 repeatedly. The second string cannot be written as there is no 2 on the table.


In this problem you are to calculate the sum of all integers from 1 to n, but you should take all powers of two with minus in the sum.For example, for n = 4 the sum is equal to  - 1 - 2 + 3 - 4 =  - 4, because 1, 2 and 4 are 20, 21 and 22 respectively.Calculate the answer for t values of n.

The first line of the input contains a single integer t (1 ≤ t ≤ 100) — the number of values of n to be processed. Each of next t lines contains a single integer n (1 ≤ n ≤ 109).

Print the requested sum for each of t integers n given in the input.

NoteThe answer for the first sample is explained in the statement.


You are given a string s and should process m queries. Each query is described by two 1-based indices li, ri and integer ki. It means that you should cyclically shift the substring s[li... ri] ki times. The queries should be processed one after another in the order they are given.One operation of a cyclic shift (rotation) is equivalent to moving the last character to the position of the first character and shifting all other characters one position to the right.For example, if the string s is abacaba and the query is l1 = 3, r1 = 6, k1 = 1 then the answer is abbacaa. If after that we would process the query l2 = 1, r2 = 4, k2 = 2 then we would get the string baabcaa.

The first line of the input contains the string s (1 ≤ |s| ≤ 10 000) in its initial state, where |s| stands for the length of s. It contains only lowercase English letters. Second line contains a single integer m (1 ≤ m ≤ 300) — the number of queries. The i-th of the next m lines contains three integers li, ri and ki (1 ≤ li ≤ ri ≤ |s|, 1 ≤ ki ≤ 1 000 000) — the description of the i-th query.

Print the resulting string s after processing all m queries.

NoteThe sample is described in problem statement.


You are given the set of vectors on the plane, each of them starting at the origin. Your task is to find a pair of vectors with the minimal non-oriented angle between them.Non-oriented angle is non-negative value, minimal between clockwise and counterclockwise direction angles. Non-oriented angle is always between 0 and π. For example, opposite directions vectors have angle equals to π.

First line of the input contains a single integer n (2 ≤ n ≤ 100 000) — the number of vectors. The i-th of the following n lines contains two integers xi and yi (|x|, |y| ≤ 10 000, x2 + y2 > 0) — the coordinates of the i-th vector. Vectors are numbered from 1 to n in order of appearing in the input. It is guaranteed that no two vectors in the input share the same direction (but they still can have opposite directions).

Print two integer numbers a and b (a ≠ b) — a pair of indices of vectors with the minimal non-oriented angle. You can print the numbers in any order. If there are many possible answers, print any.


Igor is in the museum and he wants to see as many pictures as possible.Museum can be represented as a rectangular field of n × m cells. Each cell is either empty or impassable. Empty cells are marked with '.', impassable cells are marked with '*'. Every two adjacent cells of different types (one empty and one impassable) are divided by a wall containing one picture.At the beginning Igor is in some empty cell. At every moment he can move to any empty cell that share a side with the current one.For several starting positions you should calculate the maximum number of pictures that Igor can see. Igor is able to see the picture only if he is in the cell adjacent to the wall with this picture. Igor have a lot of time, so he will examine every picture he can see.

First line of the input contains three integers n, m and k (3 ≤ n, m ≤ 1000, 1 ≤ k ≤ min(n·m, 100 000)) — the museum dimensions and the number of starting positions to process. Each of the next n lines contains m symbols '.', '*' — the description of the museum. It is guaranteed that all border cells are impassable, so Igor can't go out from the museum. Each of the last k lines contains two integers x and y (1 ≤ x ≤ n, 1 ≤ y ≤ m) — the row and the column of one of Igor's starting positions respectively. Rows are numbered from top to bottom, columns — from left to right. It is guaranteed that all starting positions are empty cells.

Print k integers — the maximum number of pictures, that Igor can see if he starts in corresponding position.


You have a rectangular chocolate bar consisting of n × m single squares. You want to eat exactly k squares, so you may need to break the chocolate bar. In one move you can break any single rectangular piece of chocolate in two rectangular pieces. You can break only by lines between squares: horizontally or vertically. The cost of breaking is equal to square of the break length.For example, if you have a chocolate bar consisting of 2 × 3 unit squares then you can break it horizontally and get two 1 × 3 pieces (the cost of such breaking is 32 = 9), or you can break it vertically in two ways and get two pieces: 2 × 1 and 2 × 2 (the cost of such breaking is 22 = 4).For several given values n, m and k find the minimum total cost of breaking. You can eat exactly k squares of chocolate if after all operations of breaking there is a set of rectangular pieces of chocolate with the total size equal to k squares. The remaining n·m - k squares are not necessarily form a single rectangular piece.

The first line of the input contains a single integer t (1 ≤ t ≤ 40910) — the number of values n, m and k to process. Each of the next t lines contains three integers n, m and k (1 ≤ n, m ≤ 30, 1 ≤ k ≤ min(n·m, 50)) — the dimensions of the chocolate bar and the number of squares you want to eat respectively.

For each n, m and k print the minimum total cost needed to break the chocolate bar, in order to make it possible to eat exactly k squares.

NoteIn the first query of the sample one needs to perform two breaks:  to split 2 × 2 bar into two pieces of 2 × 1 (cost is 22 = 4),  to split the resulting 2 × 1 into two 1 × 1 pieces (cost is 12 = 1). In the second query of the sample one wants to eat 3 unit squares. One can use exactly the same strategy as in the first query of the sample.


Given simple (without self-intersections) n-gon. It is not necessary convex. Also you are given m lines. For each line find the length of common part of the line and the n-gon.The boundary of n-gon belongs to polygon. It is possible that n-gon contains 180-degree angles.

The first line contains integers n and m (3 ≤ n ≤ 1000;1 ≤ m ≤ 100). The following n lines contain coordinates of polygon vertices (in clockwise or counterclockwise direction). All vertices are distinct. The following m lines contain line descriptions. Each of them contains two distict points of a line by their coordinates. All given in the input coordinates are real numbers, given with at most two digits after decimal point. They do not exceed 105 by absolute values.

Print m lines, the i-th line should contain the length of common part of the given n-gon and the i-th line. The answer will be considered correct if the absolute or relative error doesn't exceed 10 - 6.


Today Patrick waits for a visit from his friend Spongebob. To prepare for the visit, Patrick needs to buy some goodies in two stores located near his house. There is a d1 meter long road between his house and the first shop and a d2 meter long road between his house and the second shop. Also, there is a road of length d3 directly connecting these two shops to each other. Help Patrick calculate the minimum distance that he needs to walk in order to go to both shops and return to his house.  Patrick always starts at his house. He should visit both shops moving only along the three existing roads and return back to his house. He doesn't mind visiting the same shop or passing the same road multiple times. The only goal is to minimize the total distance traveled.

The first line of the input contains three integers d1, d2, d3 (1 ≤ d1, d2, d3 ≤ 108) — the lengths of the paths.    d1 is the length of the path connecting Patrick's house and the first shop;  d2 is the length of the path connecting Patrick's house and the second shop;  d3 is the length of the path connecting both shops.

Print the minimum distance that Patrick will have to walk in order to visit both shops and return to his house.

NoteThe first sample is shown on the picture in the problem statement. One of the optimal routes is: house  first shop  second shop  house.In the second sample one of the optimal routes is: house  first shop  house  second shop  house.