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Solve the programming task below in a Python markdown code block. You are given an array consisting of $n$ integers $a_1, a_2, \dots , a_n$ and an integer $x$. It is guaranteed that for every $i$, $1 \le a_i \le x$. Let's denote a function $f(l, r)$ which erases all values such that $l \le a_i \le r$ from the array $a$ and returns the resulting array. For example, if $a = [4, 1, 1, 4, 5, 2, 4, 3]$, then $f(2, 4) = [1, 1, 5]$. Your task is to calculate the number of pairs $(l, r)$ such that $1 \le l \le r \le x$ and $f(l, r)$ is sorted in non-descending order. Note that the empty array is also considered sorted. -----Input----- The first line contains two integers $n$ and $x$ ($1 \le n, x \le 10^6$) — the length of array $a$ and the upper limit for its elements, respectively. The second line contains $n$ integers $a_1, a_2, \dots a_n$ ($1 \le a_i \le x$). -----Output----- Print the number of pairs $1 \le l \le r \le x$ such that $f(l, r)$ is sorted in non-descending order. -----Examples----- Input 3 3 2 3 1 Output 4 Input 7 4 1 3 1 2 2 4 3 Output 6 -----Note----- In the first test case correct pairs are $(1, 1)$, $(1, 2)$, $(1, 3)$ and $(2, 3)$. In the second test case correct pairs are $(1, 3)$, $(1, 4)$, $(2, 3)$, $(2, 4)$, $(3, 3)$ and $(3, 4)$. Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. Ann has recently started commuting by subway. We know that a one ride subway ticket costs a rubles. Besides, Ann found out that she can buy a special ticket for m rides (she can buy it several times). It costs b rubles. Ann did the math; she will need to use subway n times. Help Ann, tell her what is the minimum sum of money she will have to spend to make n rides? -----Input----- The single line contains four space-separated integers n, m, a, b (1 ≤ n, m, a, b ≤ 1000) — the number of rides Ann has planned, the number of rides covered by the m ride ticket, the price of a one ride ticket and the price of an m ride ticket. -----Output----- Print a single integer — the minimum sum in rubles that Ann will need to spend. -----Examples----- Input 6 2 1 2 Output 6 Input 5 2 2 3 Output 8 -----Note----- In the first sample one of the optimal solutions is: each time buy a one ride ticket. There are other optimal solutions. For example, buy three m ride tickets. Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. Alice and Bob love playing one-dimensional battle ships. They play on the field in the form of a line consisting of n square cells (that is, on a 1 × n table). At the beginning of the game Alice puts k ships on the field without telling their positions to Bob. Each ship looks as a 1 × a rectangle (that is, it occupies a sequence of a consecutive squares of the field). The ships cannot intersect and even touch each other. After that Bob makes a sequence of "shots". He names cells of the field and Alice either says that the cell is empty ("miss"), or that the cell belongs to some ship ("hit"). But here's the problem! Alice like to cheat. May be that is why she responds to each Bob's move with a "miss". Help Bob catch Alice cheating — find Bob's first move, such that after it you can be sure that Alice cheated. -----Input----- The first line of the input contains three integers: n, k and a (1 ≤ n, k, a ≤ 2·10^5) — the size of the field, the number of the ships and the size of each ship. It is guaranteed that the n, k and a are such that you can put k ships of size a on the field, so that no two ships intersect or touch each other. The second line contains integer m (1 ≤ m ≤ n) — the number of Bob's moves. The third line contains m distinct integers x_1, x_2, ..., x_{m}, where x_{i} is the number of the cell where Bob made the i-th shot. The cells are numbered from left to right from 1 to n. -----Output----- Print a single integer — the number of such Bob's first move, after which you can be sure that Alice lied. Bob's moves are numbered from 1 to m in the order the were made. If the sought move doesn't exist, then print "-1". -----Examples----- Input 11 3 3 5 4 8 6 1 11 Output 3 Input 5 1 3 2 1 5 Output -1 Input 5 1 3 1 3 Output 1 Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. 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. -----Input----- 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 10^5 by absolute values. -----Output----- 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}. -----Examples----- Input 4 3 0 0 1 0 1 1 0 1 0 0 1 1 0 0 0 1 0 0 1 -1 Output 1.41421356237309514547 1.00000000000000000000 0.00000000000000000000 Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. Alice has a string $s$. She really likes the letter "a". She calls a string good if strictly more than half of the characters in that string are "a"s. For example "aaabb", "axaa" are good strings, and "baca", "awwwa", "" (empty string) are not. Alice can erase some characters from her string $s$. She would like to know what is the longest string remaining after erasing some characters (possibly zero) to get a good string. It is guaranteed that the string has at least one "a" in it, so the answer always exists. -----Input----- The first line contains a string $s$ ($1 \leq \|s\| \leq 50$) consisting of lowercase English letters. It is guaranteed that there is at least one "a" in $s$. -----Output----- Print a single integer, the length of the longest good string that Alice can get after erasing some characters from $s$. -----Examples----- Input xaxxxxa Output 3 Input aaabaa Output 6 -----Note----- In the first example, it's enough to erase any four of the "x"s. The answer is $3$ since that is the maximum number of characters that can remain. In the second example, we don't need to erase any characters. Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. Xenia lives in a city that has n houses built along the main ringroad. The ringroad houses are numbered 1 through n in the clockwise order. The ringroad traffic is one way and also is clockwise. Xenia has recently moved into the ringroad house number 1. As a result, she's got m things to do. In order to complete the i-th task, she needs to be in the house number a_{i} and complete all tasks with numbers less than i. Initially, Xenia is in the house number 1, find the minimum time she needs to complete all her tasks if moving from a house to a neighboring one along the ringroad takes one unit of time. -----Input----- The first line contains two integers n and m (2 ≤ n ≤ 10^5, 1 ≤ m ≤ 10^5). The second line contains m integers a_1, a_2, ..., a_{m} (1 ≤ a_{i} ≤ n). Note that Xenia can have multiple consecutive tasks in one house. -----Output----- Print a single integer — the time Xenia needs to complete all tasks. Please, do not use the %lld specifier to read or write 64-bit integers in С++. It is preferred to use the cin, cout streams or the %I64d specifier. -----Examples----- Input 4 3 3 2 3 Output 6 Input 4 3 2 3 3 Output 2 -----Note----- In the first test example the sequence of Xenia's moves along the ringroad looks as follows: 1 → 2 → 3 → 4 → 1 → 2 → 3. This is optimal sequence. So, she needs 6 time units. Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. Santa Claus likes palindromes very much. There was his birthday recently. k of his friends came to him to congratulate him, and each of them presented to him a string s_{i} having the same length n. We denote the beauty of the i-th string by a_{i}. It can happen that a_{i} is negative — that means that Santa doesn't find this string beautiful at all. Santa Claus is crazy about palindromes. He is thinking about the following question: what is the maximum possible total beauty of a palindrome which can be obtained by concatenating some (possibly all) of the strings he has? Each present can be used at most once. Note that all strings have the same length n. Recall that a palindrome is a string that doesn't change after one reverses it. Since the empty string is a palindrome too, the answer can't be negative. Even if all a_{i}'s are negative, Santa can obtain the empty string. -----Input----- The first line contains two positive integers k and n divided by space and denoting the number of Santa friends and the length of every string they've presented, respectively (1 ≤ k, n ≤ 100 000; n·k ≤ 100 000). k lines follow. The i-th of them contains the string s_{i} and its beauty a_{i} ( - 10 000 ≤ a_{i} ≤ 10 000). The string consists of n lowercase English letters, and its beauty is integer. Some of strings may coincide. Also, equal strings can have different beauties. -----Output----- In the only line print the required maximum possible beauty. -----Examples----- Input 7 3 abb 2 aaa -3 bba -1 zyz -4 abb 5 aaa 7 xyx 4 Output 12 Input 3 1 a 1 a 2 a 3 Output 6 Input 2 5 abcde 10000 abcde 10000 Output 0 -----Note----- In the first example Santa can obtain abbaaaxyxaaabba by concatenating strings 5, 2, 7, 6 and 3 (in this order). Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. Tomash keeps wandering off and getting lost while he is walking along the streets of Berland. It's no surprise! In his home town, for any pair of intersections there is exactly one way to walk from one intersection to the other one. The capital of Berland is very different! Tomash has noticed that even simple cases of ambiguity confuse him. So, when he sees a group of four distinct intersections a, b, c and d, such that there are two paths from a to c — one through b and the other one through d, he calls the group a "damn rhombus". Note that pairs (a, b), (b, c), (a, d), (d, c) should be directly connected by the roads. Schematically, a damn rhombus is shown on the figure below: [Image] Other roads between any of the intersections don't make the rhombus any more appealing to Tomash, so the four intersections remain a "damn rhombus" for him. Given that the capital of Berland has n intersections and m roads and all roads are unidirectional and are known in advance, find the number of "damn rhombi" in the city. When rhombi are compared, the order of intersections b and d doesn't matter. -----Input----- The first line of the input contains a pair of integers n, m (1 ≤ n ≤ 3000, 0 ≤ m ≤ 30000) — the number of intersections and roads, respectively. Next m lines list the roads, one per line. Each of the roads is given by a pair of integers a_{i}, b_{i} (1 ≤ a_{i}, b_{i} ≤ n;a_{i} ≠ b_{i}) — the number of the intersection it goes out from and the number of the intersection it leads to. Between a pair of intersections there is at most one road in each of the two directions. It is not guaranteed that you can get from any intersection to any other one. -----Output----- Print the required number of "damn rhombi". -----Examples----- Input 5 4 1 2 2 3 1 4 4 3 Output 1 Input 4 12 1 2 1 3 1 4 2 1 2 3 2 4 3 1 3 2 3 4 4 1 4 2 4 3 Output 12 Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. Student Valera is an undergraduate student at the University. His end of term exams are approaching and he is to pass exactly n exams. Valera is a smart guy, so he will be able to pass any exam he takes on his first try. Besides, he can take several exams on one day, and in any order. According to the schedule, a student can take the exam for the i-th subject on the day number a_{i}. However, Valera has made an arrangement with each teacher and the teacher of the i-th subject allowed him to take an exam before the schedule time on day b_{i} (b_{i} < a_{i}). Thus, Valera can take an exam for the i-th subject either on day a_{i}, or on day b_{i}. All the teachers put the record of the exam in the student's record book on the day of the actual exam and write down the date of the mark as number a_{i}. Valera believes that it would be rather strange if the entries in the record book did not go in the order of non-decreasing date. Therefore Valera asks you to help him. Find the minimum possible value of the day when Valera can take the final exam if he takes exams so that all the records in his record book go in the order of non-decreasing date. -----Input----- The first line contains a single positive integer n (1 ≤ n ≤ 5000) — the number of exams Valera will take. Each of the next n lines contains two positive space-separated integers a_{i} and b_{i} (1 ≤ b_{i} < a_{i} ≤ 10^9) — the date of the exam in the schedule and the early date of passing the i-th exam, correspondingly. -----Output----- Print a single integer — the minimum possible number of the day when Valera can take the last exam if he takes all the exams so that all the records in his record book go in the order of non-decreasing date. -----Examples----- Input 3 5 2 3 1 4 2 Output 2 Input 3 6 1 5 2 4 3 Output 6 -----Note----- In the first sample Valera first takes an exam in the second subject on the first day (the teacher writes down the schedule date that is 3). On the next day he takes an exam in the third subject (the teacher writes down the schedule date, 4), then he takes an exam in the first subject (the teacher writes down the mark with date 5). Thus, Valera takes the last exam on the second day and the dates will go in the non-decreasing order: 3, 4, 5. In the second sample Valera first takes an exam in the third subject on the fourth day. Then he takes an exam in the second subject on the fifth day. After that on the sixth day Valera takes an exam in the first subject. Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. Mike is trying rock climbing but he is awful at it. There are n holds on the wall, i-th hold is at height a_{i} off the ground. Besides, let the sequence a_{i} increase, that is, a_{i} < a_{i} + 1 for all i from 1 to n - 1; we will call such sequence a track. Mike thinks that the track a_1, ..., a_{n} has difficulty $d = \operatorname{max}_{1 \leq i \leq n - 1}(a_{i + 1} - a_{i})$. In other words, difficulty equals the maximum distance between two holds that are adjacent in height. Today Mike decided to cover the track with holds hanging on heights a_1, ..., a_{n}. To make the problem harder, Mike decided to remove one hold, that is, remove one element of the sequence (for example, if we take the sequence (1, 2, 3, 4, 5) and remove the third element from it, we obtain the sequence (1, 2, 4, 5)). However, as Mike is awful at climbing, he wants the final difficulty (i.e. the maximum difference of heights between adjacent holds after removing the hold) to be as small as possible among all possible options of removing a hold. The first and last holds must stay at their positions. Help Mike determine the minimum difficulty of the track after removing one hold. -----Input----- The first line contains a single integer n (3 ≤ n ≤ 100) — the number of holds. The next line contains n space-separated integers a_{i} (1 ≤ a_{i} ≤ 1000), where a_{i} is the height where the hold number i hangs. The sequence a_{i} is increasing (i.e. each element except for the first one is strictly larger than the previous one). -----Output----- Print a single number — the minimum difficulty of the track after removing a single hold. -----Examples----- Input 3 1 4 6 Output 5 Input 5 1 2 3 4 5 Output 2 Input 5 1 2 3 7 8 Output 4 -----Note----- In the first sample you can remove only the second hold, then the sequence looks like (1, 6), the maximum difference of the neighboring elements equals 5. In the second test after removing every hold the difficulty equals 2. In the third test you can obtain sequences (1, 3, 7, 8), (1, 2, 7, 8), (1, 2, 3, 8), for which the difficulty is 4, 5 and 5, respectively. Thus, after removing the second element we obtain the optimal answer — 4. Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. Завтра у хоккейной команды, которой руководит Евгений, важный матч. Евгению нужно выбрать шесть игроков, которые выйдут на лед в стартовом составе: один вратарь, два защитника и три нападающих. Так как это стартовый состав, Евгения больше волнует, насколько красива будет команда на льду, чем способности игроков. А именно, Евгений хочет выбрать такой стартовый состав, чтобы номера любых двух игроков из стартового состава отличались не более, чем в два раза. Например, игроки с номерами 13, 14, 10, 18, 15 и 20 устроят Евгения, а если, например, на лед выйдут игроки с номерами 8 и 17, то это не устроит Евгения. Про каждого из игроков вам известно, на какой позиции он играет (вратарь, защитник или нападающий), а также его номер. В хоккее номера игроков не обязательно идут подряд. Посчитайте число различных стартовых составов из одного вратаря, двух защитников и трех нападающих, которые может выбрать Евгений, чтобы выполнялось его условие красоты. -----Входные данные----- Первая строка содержит три целых числа g, d и f (1 ≤ g ≤ 1 000, 1 ≤ d ≤ 1 000, 1 ≤ f ≤ 1 000) — число вратарей, защитников и нападающих в команде Евгения. Вторая строка содержит g целых чисел, каждое в пределах от 1 до 100 000 — номера вратарей. Третья строка содержит d целых чисел, каждое в пределах от 1 до 100 000 — номера защитников. Четвертая строка содержит f целых чисел, каждое в пределах от 1 до 100 000 — номера нападающих. Гарантируется, что общее количество игроков не превосходит 1 000, т. е. g + d + f ≤ 1 000. Все g + d + f номеров игроков различны. -----Выходные данные----- Выведите одно целое число — количество возможных стартовых составов. -----Примеры----- Входные данные 1 2 3 15 10 19 20 11 13 Выходные данные 1 Входные данные 2 3 4 16 40 20 12 19 13 21 11 10 Выходные данные 6 -----Примечание----- В первом примере всего один вариант для выбора состава, который удовлетворяет описанным условиям, поэтому ответ 1. Во втором примере подходят следующие игровые сочетания (в порядке вратарь-защитник-защитник-нападающий-нападающий-нападающий): 16 20 12 13 21 11 16 20 12 13 11 10 16 20 19 13 21 11 16 20 19 13 11 10 16 12 19 13 21 11 16 12 19 13 11 10 Таким образом, ответ на этот пример — 6. Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. Arkady invited Anna for a dinner to a sushi restaurant. The restaurant is a bit unusual: it offers $n$ pieces of sushi aligned in a row, and a customer has to choose a continuous subsegment of these sushi to buy. The pieces of sushi are of two types: either with tuna or with eel. Let's denote the type of the $i$-th from the left sushi as $t_i$, where $t_i = 1$ means it is with tuna, and $t_i = 2$ means it is with eel. Arkady does not like tuna, Anna does not like eel. Arkady wants to choose such a continuous subsegment of sushi that it has equal number of sushi of each type and each half of the subsegment has only sushi of one type. For example, subsegment $[2, 2, 2, 1, 1, 1]$ is valid, but subsegment $[1, 2, 1, 2, 1, 2]$ is not, because both halves contain both types of sushi. Find the length of the longest continuous subsegment of sushi Arkady can buy. -----Input----- The first line contains a single integer $n$ ($2 \le n \le 100\,000$) — the number of pieces of sushi. The second line contains $n$ integers $t_1$, $t_2$, ..., $t_n$ ($t_i = 1$, denoting a sushi with tuna or $t_i = 2$, denoting a sushi with eel), representing the types of sushi from left to right. It is guaranteed that there is at least one piece of sushi of each type. Note that it means that there is at least one valid continuous segment. -----Output----- Print a single integer — the maximum length of a valid continuous segment. -----Examples----- Input 7 2 2 2 1 1 2 2 Output 4 Input 6 1 2 1 2 1 2 Output 2 Input 9 2 2 1 1 1 2 2 2 2 Output 6 -----Note----- In the first example Arkady can choose the subsegment $[2, 2, 1, 1]$ or the subsegment $[1, 1, 2, 2]$ with length $4$. In the second example there is no way but to choose one of the subsegments $[2, 1]$ or $[1, 2]$ with length $2$. In the third example Arkady's best choice is the subsegment $[1, 1, 1, 2, 2, 2]$. Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. Ostap already settled down in Rio de Janiero suburb and started to grow a tree in his garden. Recall that a tree is a connected undirected acyclic graph. Ostap's tree now has n vertices. He wants to paint some vertices of the tree black such that from any vertex u there is at least one black vertex v at distance no more than k. Distance between two vertices of the tree is the minimum possible number of edges of the path between them. As this number of ways to paint the tree can be large, Ostap wants you to compute it modulo 10^9 + 7. Two ways to paint the tree are considered different if there exists a vertex that is painted black in one way and is not painted in the other one. -----Input----- The first line of the input contains two integers n and k (1 ≤ n ≤ 100, 0 ≤ k ≤ min(20, n - 1)) — the number of vertices in Ostap's tree and the maximum allowed distance to the nearest black vertex. Don't miss the unusual constraint for k. Each of the next n - 1 lines contain two integers u_{i} and v_{i} (1 ≤ u_{i}, v_{i} ≤ n) — indices of vertices, connected by the i-th edge. It's guaranteed that given graph is a tree. -----Output----- Print one integer — the remainder of division of the number of ways to paint the tree by 1 000 000 007 (10^9 + 7). -----Examples----- Input 2 0 1 2 Output 1 Input 2 1 1 2 Output 3 Input 4 1 1 2 2 3 3 4 Output 9 Input 7 2 1 2 2 3 1 4 4 5 1 6 6 7 Output 91 -----Note----- In the first sample, Ostap has to paint both vertices black. In the second sample, it is enough to paint only one of two vertices, thus the answer is 3: Ostap can paint only vertex 1, only vertex 2, vertices 1 and 2 both. In the third sample, the valid ways to paint vertices are: {1, 3}, {1, 4}, {2, 3}, {2, 4}, {1, 2, 3}, {1, 2, 4}, {1, 3, 4}, {2, 3, 4}, {1, 2, 3, 4}. Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. Sereja and his friends went to a picnic. The guys had n soda bottles just for it. Sereja forgot the bottle opener as usual, so the guys had to come up with another way to open bottles. Sereja knows that the i-th bottle is from brand a_{i}, besides, you can use it to open other bottles of brand b_{i}. You can use one bottle to open multiple other bottles. Sereja can open bottle with opened bottle or closed bottle. Knowing this, Sereja wants to find out the number of bottles they've got that they won't be able to open in any way. Help him and find this number. -----Input----- The first line contains integer n (1 ≤ n ≤ 100) — the number of bottles. The next n lines contain the bottles' description. The i-th line contains two integers a_{i}, b_{i} (1 ≤ a_{i}, b_{i} ≤ 1000) — the description of the i-th bottle. -----Output----- In a single line print a single integer — the answer to the problem. -----Examples----- Input 4 1 1 2 2 3 3 4 4 Output 4 Input 4 1 2 2 3 3 4 4 1 Output 0 Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. You are given N items. The value of the i-th item (1 \leq i \leq N) is v_i. Your have to select at least A and at most B of these items. Under this condition, find the maximum possible arithmetic mean of the values of selected items. Additionally, find the number of ways to select items so that the mean of the values of selected items is maximized. -----Constraints----- - 1 \leq N \leq 50 - 1 \leq A,B \leq N - 1 \leq v_i \leq 10^{15} - Each v_i is an integer. -----Input----- The input is given from Standard Input in the following format: N A B v_1 v_2 ... v_N -----Output----- Print two lines. The first line should contain the maximum possible arithmetic mean of the values of selected items. The output should be considered correct if the absolute or relative error is at most 10^{-6}. The second line should contain the number of ways to select items so that the mean of the values of selected items is maximized. -----Sample Input----- 5 2 2 1 2 3 4 5 -----Sample Output----- 4.500000 1 The mean of the values of selected items will be maximized when selecting the fourth and fifth items. Hence, the first line of the output should contain 4.5. There is no other way to select items so that the mean of the values will be 4.5, and thus the second line of the output should contain 1. Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. Given are N points (x_i, y_i) in a two-dimensional plane. Find the minimum radius of a circle such that all the points are inside or on it. -----Constraints----- - 2 \leq N \leq 50 - 0 \leq x_i \leq 1000 - 0 \leq y_i \leq 1000 - The given N points are all different. - The values in input are all integers. -----Input----- Input is given from Standard Input in the following format: N x_1 y_1 : x_N y_N -----Output----- Print the minimum radius of a circle such that all the N points are inside or on it. Your output will be considered correct if the absolute or relative error from our answer is at most 10^{-6}. -----Sample Input----- 2 0 0 1 0 -----Sample Output----- 0.500000000000000000 Both points are contained in the circle centered at (0.5,0) with a radius of 0.5. Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. Given is an integer S. Find how many sequences there are whose terms are all integers greater than or equal to 3, and whose sum is equal to S. The answer can be very large, so output it modulo 10^9 + 7. -----Constraints----- - 1 \leq S \leq 2000 - All values in input are integers. -----Input----- Input is given from Standard Input in the following format: S -----Output----- Print the answer. -----Sample Input----- 7 -----Sample Output----- 3 3 sequences satisfy the condition: \{3,4\}, \{4,3\} and \{7\}. Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. Ibis is fighting with a monster. The health of the monster is H. Ibis can cast N kinds of spells. Casting the i-th spell decreases the monster's health by A_i, at the cost of B_i Magic Points. The same spell can be cast multiple times. There is no way other than spells to decrease the monster's health. Ibis wins when the health of the monster becomes 0 or below. Find the minimum total Magic Points that have to be consumed before winning. -----Constraints----- - 1 \leq H \leq 10^4 - 1 \leq N \leq 10^3 - 1 \leq A_i \leq 10^4 - 1 \leq B_i \leq 10^4 - All values in input are integers. -----Input----- Input is given from Standard Input in the following format: H N A_1 B_1 : A_N B_N -----Output----- Print the minimum total Magic Points that have to be consumed before winning. -----Sample Input----- 9 3 8 3 4 2 2 1 -----Sample Output----- 4 First, let us cast the first spell to decrease the monster's health by 8, at the cost of 3 Magic Points. The monster's health is now 1. Then, cast the third spell to decrease the monster's health by 2, at the cost of 1 Magic Point. The monster's health is now -1. In this way, we can win at the total cost of 4 Magic Points. Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. We have N+1 integers: 10^{100}, 10^{100}+1, ..., 10^{100}+N. We will choose K or more of these integers. Find the number of possible values of the sum of the chosen numbers, modulo (10^9+7). -----Constraints----- - 1 \leq N \leq 2\times 10^5 - 1 \leq K \leq N+1 - All values in input are integers. -----Input----- Input is given from Standard Input in the following format: N K -----Output----- Print the number of possible values of the sum, modulo (10^9+7). -----Sample Input----- 3 2 -----Sample Output----- 10 The sum can take 10 values, as follows: - (10^{100})+(10^{100}+1)=2\times 10^{100}+1 - (10^{100})+(10^{100}+2)=2\times 10^{100}+2 - (10^{100})+(10^{100}+3)=(10^{100}+1)+(10^{100}+2)=2\times 10^{100}+3 - (10^{100}+1)+(10^{100}+3)=2\times 10^{100}+4 - (10^{100}+2)+(10^{100}+3)=2\times 10^{100}+5 - (10^{100})+(10^{100}+1)+(10^{100}+2)=3\times 10^{100}+3 - (10^{100})+(10^{100}+1)+(10^{100}+3)=3\times 10^{100}+4 - (10^{100})+(10^{100}+2)+(10^{100}+3)=3\times 10^{100}+5 - (10^{100}+1)+(10^{100}+2)+(10^{100}+3)=3\times 10^{100}+6 - (10^{100})+(10^{100}+1)+(10^{100}+2)+(10^{100}+3)=4\times 10^{100}+6 Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. You've got array a[1], a[2], ..., a[n], consisting of n integers. Count the number of ways to split all the elements of the array into three contiguous parts so that the sum of elements in each part is the same. More formally, you need to find the number of such pairs of indices i, j (2 ≤ i ≤ j ≤ n - 1), that $\sum_{k = 1}^{i - 1} a_{k} = \sum_{k = i}^{j} a_{k} = \sum_{k = j + 1}^{n} a_{k}$. -----Input----- The first line contains integer n (1 ≤ n ≤ 5·10^5), showing how many numbers are in the array. The second line contains n integers a[1], a[2], ..., a[n] (\|a[i]\| ≤ 10^9) — the elements of array a. -----Output----- Print a single integer — the number of ways to split the array into three parts with the same sum. -----Examples----- Input 5 1 2 3 0 3 Output 2 Input 4 0 1 -1 0 Output 1 Input 2 4 1 Output 0 Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. Sasha and Dima want to buy two $n$-tier cakes. Each cake should consist of $n$ different tiers: from the size of $1$ to the size of $n$. Tiers should go in order from the smallest to the biggest (from top to bottom). They live on the same street, there are $2 \cdot n$ houses in a row from left to right. Each house has a pastry shop where you can buy a cake tier. Unfortunately, in each pastry shop you can buy only one tier of only one specific size: in the $i$-th house you can buy a tier of the size $a_i$ ($1 \le a_i \le n$). Since the guys carry already purchased tiers, and it is impossible to insert a new tier in the middle of the cake, they agreed to buy tiers from the smallest to the biggest. That is, each of them buys tiers in order: $1$, then $2$, then $3$ and so on up to $n$. Initially, Sasha and Dima are located near the first (leftmost) house. Output the minimum distance that they will have to walk in total to buy both cakes. The distance between any two neighboring houses is exactly $1$. -----Input----- The first line of the input contains an integer number $n$ — the number of tiers in each cake ($1 \le n \le 10^5$). The second line contains $2 \cdot n$ integers $a_1, a_2, \dots, a_{2n}$ ($1 \le a_i \le n$), where $a_i$ is equal to the size of the tier, which can be bought in the $i$-th house. Remember that in each house you can buy only one tier. It is guaranteed that every number from $1$ to $n$ occurs in $a$ exactly two times. -----Output----- Print one number — the minimum distance that the guys have to walk in total to buy both cakes. Guys can be near same house at the same time. They begin near the first (leftmost) house. Each of the guys should buy $n$ tiers in ascending order of their sizes. -----Examples----- Input 3 1 1 2 2 3 3 Output 9 Input 2 2 1 1 2 Output 5 Input 4 4 1 3 2 2 3 1 4 Output 17 -----Note----- In the first example, the possible optimal sequence of actions is: Sasha buys a tier of size $1$ near the $1$-st house ($a_1=1$); Dima goes to the house $2$; Dima buys a tier of size $1$ near the $2$-nd house ($a_2=1$); Sasha goes to the house $4$; Sasha buys a tier of size $2$ near the $4$-th house ($a_4=2$); Sasha goes to the house $5$; Sasha buys a tier of size $3$ near the $5$-th house ($a_5=3$); Dima goes to the house $3$; Dima buys a tier of size $2$ near the $3$-rd house ($a_3=2$); Dima goes to the house $6$; Dima buys a tier of size $3$ near the $6$-th house ($a_6=3$). So, Sasha goes the distance $3+1=4$, and Dima goes the distance $1+1+3=5$. In total, they cover a distance of $4+5=9$. You can make sure that with any other sequence of actions they will walk no less distance. Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. There are $n$ pillars aligned in a row and numbered from $1$ to $n$. Initially each pillar contains exactly one disk. The $i$-th pillar contains a disk having radius $a_i$. You can move these disks from one pillar to another. You can take a disk from pillar $i$ and place it on top of pillar $j$ if all these conditions are met: there is no other pillar between pillars $i$ and $j$. Formally, it means that $\|i - j\| = 1$; pillar $i$ contains exactly one disk; either pillar $j$ contains no disks, or the topmost disk on pillar $j$ has radius strictly greater than the radius of the disk you move. When you place a disk on a pillar that already has some disks on it, you put the new disk on top of previously placed disks, so the new disk will be used to check the third condition if you try to place another disk on the same pillar. You may take any disk and place it on other pillar any number of times, provided that every time you do it, all three aforementioned conditions are met. Now you wonder, is it possible to place all $n$ disks on the same pillar simultaneously? -----Input----- The first line contains one integer $n$ ($3 \le n \le 2 \cdot 10^5$) — the number of pillars. The second line contains $n$ integers $a_1$, $a_2$, ..., $a_i$ ($1 \le a_i \le n$), where $a_i$ is the radius of the disk initially placed on the $i$-th pillar. All numbers $a_i$ are distinct. -----Output----- Print YES if it is possible to place all the disks on the same pillar simultaneously, and NO otherwise. You may print each letter in any case (YES, yes, Yes will all be recognized as positive answer, NO, no and nO will all be recognized as negative answer). -----Examples----- Input 4 1 3 4 2 Output YES Input 3 3 1 2 Output NO -----Note----- In the first case it is possible to place all disks on pillar $3$ using the following sequence of actions: take the disk with radius $3$ from pillar $2$ and place it on top of pillar $3$; take the disk with radius $1$ from pillar $1$ and place it on top of pillar $2$; take the disk with radius $2$ from pillar $4$ and place it on top of pillar $3$; take the disk with radius $1$ from pillar $2$ and place it on top of pillar $3$. Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. Bill is a famous mathematician in BubbleLand. Thanks to his revolutionary math discoveries he was able to make enough money to build a beautiful house. Unfortunately, for not paying property tax on time, court decided to punish Bill by making him lose a part of his property. Bill’s property can be observed as a convex regular 2n-sided polygon A_0 A_1... A_2n - 1 A_2n, A_2n = A_0, with sides of the exactly 1 meter in length. Court rules for removing part of his property are as follows: Split every edge A_{k} A_{k} + 1, k = 0... 2n - 1 in n equal parts of size 1 / n with points P_0, P_1, ..., P_{n} - 1 On every edge A_2k A_2k + 1, k = 0... n - 1 court will choose one point B_2k = P_{i} for some i = 0, ..., n - 1 such that $\cup_{i = 0}^{n - 1} B_{2i} = \cup_{i = 0}^{n - 1} P_{i}$ On every edge A_2k + 1A_2k + 2, k = 0...n - 1 Bill will choose one point B_2k + 1 = P_{i} for some i = 0, ..., n - 1 such that $\cup_{i = 0}^{n - 1} B_{2 i + 1} = \cup_{i = 0}^{n - 1} P_{i}$ Bill gets to keep property inside of 2n-sided polygon B_0 B_1... B_2n - 1 Luckily, Bill found out which B_2k points the court chose. Even though he is a great mathematician, his house is very big and he has a hard time calculating. Therefore, he is asking you to help him choose points so he maximizes area of property he can keep. -----Input----- The first line contains one integer number n (2 ≤ n ≤ 50000), representing number of edges of 2n-sided polygon. The second line contains n distinct integer numbers B_2k (0 ≤ B_2k ≤ n - 1, k = 0... n - 1) separated by a single space, representing points the court chose. If B_2k = i, the court chose point P_{i} on side A_2k A_2k + 1. -----Output----- Output contains n distinct integers separated by a single space representing points B_1, B_3, ..., B_2n - 1 Bill should choose in order to maximize the property area. If there are multiple solutions that maximize the area, return any of them. -----Example----- Input 3 0 1 2 Output 0 2 1 -----Note----- To maximize area Bill should choose points: B_1 = P_0, B_3 = P_2, B_5 = P_1 [Image] Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. Recently Monocarp got a job. His working day lasts exactly $m$ minutes. During work, Monocarp wants to drink coffee at certain moments: there are $n$ minutes $a_1, a_2, \dots, a_n$, when he is able and willing to take a coffee break (for the sake of simplicity let's consider that each coffee break lasts exactly one minute). However, Monocarp's boss doesn't like when Monocarp takes his coffee breaks too often. So for the given coffee break that is going to be on minute $a_i$, Monocarp must choose the day in which he will drink coffee during the said minute, so that every day at least $d$ minutes pass between any two coffee breaks. Monocarp also wants to take these $n$ coffee breaks in a minimum possible number of working days (he doesn't count days when he is not at work, and he doesn't take coffee breaks on such days). Take into account that more than $d$ minutes pass between the end of any working day and the start of the following working day. For each of the $n$ given minutes determine the day, during which Monocarp should take a coffee break in this minute. You have to minimize the number of days spent. -----Input----- The first line contains three integers $n$, $m$, $d$ $(1 \le n \le 2\cdot10^{5}, n \le m \le 10^{9}, 1 \le d \le m)$ — the number of coffee breaks Monocarp wants to have, the length of each working day, and the minimum number of minutes between any two consecutive coffee breaks. The second line contains $n$ distinct integers $a_1, a_2, \dots, a_n$ $(1 \le a_i \le m)$, where $a_i$ is some minute when Monocarp wants to have a coffee break. -----Output----- In the first line, write the minimum number of days required to make a coffee break in each of the $n$ given minutes. In the second line, print $n$ space separated integers. The $i$-th of integers should be the index of the day during which Monocarp should have a coffee break at minute $a_i$. Days are numbered from $1$. If there are multiple optimal solutions, you may print any of them. -----Examples----- Input 4 5 3 3 5 1 2 Output 3 3 1 1 2 Input 10 10 1 10 5 7 4 6 3 2 1 9 8 Output 2 2 1 1 2 2 1 2 1 1 2 -----Note----- In the first example, Monocarp can take two coffee breaks during the first day (during minutes $1$ and $5$, $3$ minutes will pass between these breaks). One break during the second day (at minute $2$), and one break during the third day (at minute $3$). In the second example, Monocarp can determine the day of the break as follows: if the minute when he wants to take a break is odd, then this break is on the first day, if it is even, then this break is on the second day. Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. The Queen of England has n trees growing in a row in her garden. At that, the i-th (1 ≤ i ≤ n) tree from the left has height a_{i} meters. Today the Queen decided to update the scenery of her garden. She wants the trees' heights to meet the condition: for all i (1 ≤ i < n), a_{i} + 1 - a_{i} = k, where k is the number the Queen chose. Unfortunately, the royal gardener is not a machine and he cannot fulfill the desire of the Queen instantly! In one minute, the gardener can either decrease the height of a tree to any positive integer height or increase the height of a tree to any positive integer height. How should the royal gardener act to fulfill a whim of Her Majesty in the minimum number of minutes? -----Input----- The first line contains two space-separated integers: n, k (1 ≤ n, k ≤ 1000). The second line contains n space-separated integers a_1, a_2, ..., a_{n} (1 ≤ a_{i} ≤ 1000) — the heights of the trees in the row. -----Output----- In the first line print a single integer p — the minimum number of minutes the gardener needs. In the next p lines print the description of his actions. If the gardener needs to increase the height of the j-th (1 ≤ j ≤ n) tree from the left by x (x ≥ 1) meters, then print in the corresponding line "+ j x". If the gardener needs to decrease the height of the j-th (1 ≤ j ≤ n) tree from the left by x (x ≥ 1) meters, print on the corresponding line "- j x". If there are multiple ways to make a row of trees beautiful in the minimum number of actions, you are allowed to print any of them. -----Examples----- Input 4 1 1 2 1 5 Output 2 + 3 2 - 4 1 Input 4 1 1 2 3 4 Output 0 Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. To make a paper airplane, one has to use a rectangular piece of paper. From a sheet of standard size you can make $s$ airplanes. A group of $k$ people decided to make $n$ airplanes each. They are going to buy several packs of paper, each of them containing $p$ sheets, and then distribute the sheets between the people. Each person should have enough sheets to make $n$ airplanes. How many packs should they buy? -----Input----- The only line contains four integers $k$, $n$, $s$, $p$ ($1 \le k, n, s, p \le 10^4$) — the number of people, the number of airplanes each should make, the number of airplanes that can be made using one sheet and the number of sheets in one pack, respectively. -----Output----- Print a single integer — the minimum number of packs they should buy. -----Examples----- Input 5 3 2 3 Output 4 Input 5 3 100 1 Output 5 -----Note----- In the first sample they have to buy $4$ packs of paper: there will be $12$ sheets in total, and giving $2$ sheets to each person is enough to suit everyone's needs. In the second sample they have to buy a pack for each person as they can't share sheets. Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. After Vitaly was expelled from the university, he became interested in the graph theory. Vitaly especially liked the cycles of an odd length in which each vertex occurs at most once. Vitaly was wondering how to solve the following problem. You are given an undirected graph consisting of n vertices and m edges, not necessarily connected, without parallel edges and loops. You need to find t — the minimum number of edges that must be added to the given graph in order to form a simple cycle of an odd length, consisting of more than one vertex. Moreover, he must find w — the number of ways to add t edges in order to form a cycle of an odd length (consisting of more than one vertex). It is prohibited to add loops or parallel edges. Two ways to add edges to the graph are considered equal if they have the same sets of added edges. Since Vitaly does not study at the university, he asked you to help him with this task. -----Input----- The first line of the input contains two integers n and m ($3 \leq n \leq 10^{5}, 0 \leq m \leq \operatorname{min}(\frac{n(n - 1)}{2}, 10^{5})$ — the number of vertices in the graph and the number of edges in the graph. Next m lines contain the descriptions of the edges of the graph, one edge per line. Each edge is given by a pair of integers a_{i}, b_{i} (1 ≤ a_{i}, b_{i} ≤ n) — the vertices that are connected by the i-th edge. All numbers in the lines are separated by a single space. It is guaranteed that the given graph doesn't contain any loops and parallel edges. The graph isn't necessarily connected. -----Output----- Print in the first line of the output two space-separated integers t and w — the minimum number of edges that should be added to the graph to form a simple cycle of an odd length consisting of more than one vertex where each vertex occurs at most once, and the number of ways to do this. -----Examples----- Input 4 4 1 2 1 3 4 2 4 3 Output 1 2 Input 3 3 1 2 2 3 3 1 Output 0 1 Input 3 0 Output 3 1 -----Note----- The simple cycle is a cycle that doesn't contain any vertex twice. Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. You are given a positive integer $m$ and two integer sequence: $a=[a_1, a_2, \ldots, a_n]$ and $b=[b_1, b_2, \ldots, b_n]$. Both of these sequence have a length $n$. Permutation is a sequence of $n$ different positive integers from $1$ to $n$. For example, these sequences are permutations: $[1]$, $[1,2]$, $[2,1]$, $[6,7,3,4,1,2,5]$. These are not: $[0]$, $[1,1]$, $[2,3]$. You need to find the non-negative integer $x$, and increase all elements of $a_i$ by $x$, modulo $m$ (i.e. you want to change $a_i$ to $(a_i + x) \bmod m$), so it would be possible to rearrange elements of $a$ to make it equal $b$, among them you need to find the smallest possible $x$. In other words, you need to find the smallest non-negative integer $x$, for which it is possible to find some permutation $p=[p_1, p_2, \ldots, p_n]$, such that for all $1 \leq i \leq n$, $(a_i + x) \bmod m = b_{p_i}$, where $y \bmod m$ — remainder of division of $y$ by $m$. For example, if $m=3$, $a = [0, 0, 2, 1], b = [2, 0, 1, 1]$, you can choose $x=1$, and $a$ will be equal to $[1, 1, 0, 2]$ and you can rearrange it to make it equal $[2, 0, 1, 1]$, which is equal to $b$. -----Input----- The first line contains two integers $n,m$ ($1 \leq n \leq 2000, 1 \leq m \leq 10^9$): number of elemens in arrays and $m$. The second line contains $n$ integers $a_1, a_2, \ldots, a_n$ ($0 \leq a_i < m$). The third line contains $n$ integers $b_1, b_2, \ldots, b_n$ ($0 \leq b_i < m$). It is guaranteed that there exists some non-negative integer $x$, such that it would be possible to find some permutation $p_1, p_2, \ldots, p_n$ such that $(a_i + x) \bmod m = b_{p_i}$. -----Output----- Print one integer, the smallest non-negative integer $x$, such that it would be possible to find some permutation $p_1, p_2, \ldots, p_n$ such that $(a_i + x) \bmod m = b_{p_i}$ for all $1 \leq i \leq n$. -----Examples----- Input 4 3 0 0 2 1 2 0 1 1 Output 1 Input 3 2 0 0 0 1 1 1 Output 1 Input 5 10 0 0 0 1 2 2 1 0 0 0 Output 0 Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. Hideo Kojima has just quit his job at Konami. Now he is going to find a new place to work. Despite being such a well-known person, he still needs a CV to apply for a job. During all his career Hideo has produced n games. Some of them were successful, some were not. Hideo wants to remove several of them (possibly zero) from his CV to make a better impression on employers. As a result there should be no unsuccessful game which comes right after successful one in his CV. More formally, you are given an array s_1, s_2, ..., s_{n} of zeros and ones. Zero corresponds to an unsuccessful game, one — to a successful one. Games are given in order they were produced, and Hideo can't swap these values. He should remove some elements from this array in such a way that no zero comes right after one. Besides that, Hideo still wants to mention as much games in his CV as possible. Help this genius of a man determine the maximum number of games he can leave in his CV. -----Input----- The first line contains one integer number n (1 ≤ n ≤ 100). The second line contains n space-separated integer numbers s_1, s_2, ..., s_{n} (0 ≤ s_{i} ≤ 1). 0 corresponds to an unsuccessful game, 1 — to a successful one. -----Output----- Print one integer — the maximum number of games Hideo can leave in his CV so that no unsuccessful game comes after a successful one. -----Examples----- Input 4 1 1 0 1 Output 3 Input 6 0 1 0 0 1 0 Output 4 Input 1 0 Output 1 Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. The problem describes the properties of a command line. The description somehow resembles the one you usually see in real operating systems. However, there are differences in the behavior. Please make sure you've read the statement attentively and use it as a formal document. In the Pindows operating system a strings are the lexemes of the command line — the first of them is understood as the name of the program to run and the following lexemes are its arguments. For example, as we execute the command " run.exe one, two . ", we give four lexemes to the Pindows command line: "run.exe", "one,", "two", ".". More formally, if we run a command that can be represented as string s (that has no quotes), then the command line lexemes are maximal by inclusion substrings of string s that contain no spaces. To send a string with spaces or an empty string as a command line lexeme, we can use double quotes. The block of characters that should be considered as one lexeme goes inside the quotes. Embedded quotes are prohibited — that is, for each occurrence of character """ we should be able to say clearly that the quotes are opening or closing. For example, as we run the command ""run.exe o" "" " ne, " two . " " ", we give six lexemes to the Pindows command line: "run.exe o", "" (an empty string), " ne, ", "two", ".", " " (a single space). It is guaranteed that each lexeme of the command line is either surrounded by spaces on both sides or touches the corresponding command border. One of its consequences is: the opening brackets are either the first character of the string or there is a space to the left of them. You have a string that consists of uppercase and lowercase English letters, digits, characters ".,?!"" and spaces. It is guaranteed that this string is a correct OS Pindows command line string. Print all lexemes of this command line string. Consider the character """ to be used only in order to denote a single block of characters into one command line lexeme. In particular, the consequence is that the given string has got an even number of such characters. -----Input----- The single line contains a non-empty string s. String s consists of at most 10^5 characters. Each character is either an uppercase or a lowercase English letter, or a digit, or one of the ".,?!"" signs, or a space. It is guaranteed that the given string is some correct command line string of the OS Pindows. It is guaranteed that the given command line string contains at least one lexeme. -----Output----- In the first line print the first lexeme, in the second line print the second one and so on. To make the output clearer, print the "<" (less) character to the left of your lexemes and the ">" (more) character to the right. Print the lexemes in the order in which they occur in the command. Please, follow the given output format strictly. For more clarifications on the output format see the test samples. -----Examples----- Input "RUn.exe O" "" " 2ne, " two! . " " Output <RUn.exe O> <> < 2ne, > <two!> <.> < > Input firstarg second "" Output <firstarg> <second> <> Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. Bob is decorating his kitchen, more precisely, the floor. He has found a prime candidate for the tiles he will use. They come in a simple form factor — a square tile that is diagonally split into white and black part as depicted in the figure below. [Image] The dimension of this tile is perfect for this kitchen, as he will need exactly $w \times h$ tiles without any scraps. That is, the width of the kitchen is $w$ tiles, and the height is $h$ tiles. As each tile can be rotated in one of four ways, he still needs to decide on how exactly he will tile the floor. There is a single aesthetic criterion that he wants to fulfil: two adjacent tiles must not share a colour on the edge — i.e. one of the tiles must have a white colour on the shared border, and the second one must be black. [Image] The picture on the left shows one valid tiling of a $3 \times 2$ kitchen. The picture on the right shows an invalid arrangement, as the bottom two tiles touch with their white parts. Find the number of possible tilings. As this number may be large, output its remainder when divided by $998244353$ (a prime number). -----Input----- The only line contains two space separated integers $w$, $h$ ($1 \leq w,h \leq 1\,000$) — the width and height of the kitchen, measured in tiles. -----Output----- Output a single integer $n$ — the remainder of the number of tilings when divided by $998244353$. -----Examples----- Input 2 2 Output 16 Input 2 4 Output 64 Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. New Year is coming in Line World! In this world, there are n cells numbered by integers from 1 to n, as a 1 × n board. People live in cells. However, it was hard to move between distinct cells, because of the difficulty of escaping the cell. People wanted to meet people who live in other cells. So, user tncks0121 has made a transportation system to move between these cells, to celebrate the New Year. First, he thought of n - 1 positive integers a_1, a_2, ..., a_{n} - 1. For every integer i where 1 ≤ i ≤ n - 1 the condition 1 ≤ a_{i} ≤ n - i holds. Next, he made n - 1 portals, numbered by integers from 1 to n - 1. The i-th (1 ≤ i ≤ n - 1) portal connects cell i and cell (i + a_{i}), and one can travel from cell i to cell (i + a_{i}) using the i-th portal. Unfortunately, one cannot use the portal backwards, which means one cannot move from cell (i + a_{i}) to cell i using the i-th portal. It is easy to see that because of condition 1 ≤ a_{i} ≤ n - i one can't leave the Line World using portals. Currently, I am standing at cell 1, and I want to go to cell t. However, I don't know whether it is possible to go there. Please determine whether I can go to cell t by only using the construted transportation system. -----Input----- The first line contains two space-separated integers n (3 ≤ n ≤ 3 × 10^4) and t (2 ≤ t ≤ n) — the number of cells, and the index of the cell which I want to go to. The second line contains n - 1 space-separated integers a_1, a_2, ..., a_{n} - 1 (1 ≤ a_{i} ≤ n - i). It is guaranteed, that using the given transportation system, one cannot leave the Line World. -----Output----- If I can go to cell t using the transportation system, print "YES". Otherwise, print "NO". -----Examples----- Input 8 4 1 2 1 2 1 2 1 Output YES Input 8 5 1 2 1 2 1 1 1 Output NO -----Note----- In the first sample, the visited cells are: 1, 2, 4; so we can successfully visit the cell 4. In the second sample, the possible cells to visit are: 1, 2, 4, 6, 7, 8; so we can't visit the cell 5, which we want to visit. Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. Ashish has a tree consisting of $n$ nodes numbered $1$ to $n$ rooted at node $1$. The $i$-th node in the tree has a cost $a_i$, and binary digit $b_i$ is written in it. He wants to have binary digit $c_i$ written in the $i$-th node in the end. To achieve this, he can perform the following operation any number of times: Select any $k$ nodes from the subtree of any node $u$, and shuffle the digits in these nodes as he wishes, incurring a cost of $k \cdot a_u$. Here, he can choose $k$ ranging from $1$ to the size of the subtree of $u$. He wants to perform the operations in such a way that every node finally has the digit corresponding to its target. Help him find the minimum total cost he needs to spend so that after all the operations, every node $u$ has digit $c_u$ written in it, or determine that it is impossible. -----Input----- First line contains a single integer $n$ $(1 \le n \le 2 \cdot 10^5)$ denoting the number of nodes in the tree. $i$-th line of the next $n$ lines contains 3 space-separated integers $a_i$, $b_i$, $c_i$ $(1 \leq a_i \leq 10^9, 0 \leq b_i, c_i \leq 1)$ — the cost of the $i$-th node, its initial digit and its goal digit. Each of the next $n - 1$ lines contain two integers $u$, $v$ $(1 \leq u, v \leq n, \text{ } u \ne v)$, meaning that there is an edge between nodes $u$ and $v$ in the tree. -----Output----- Print the minimum total cost to make every node reach its target digit, and $-1$ if it is impossible. -----Examples----- Input 5 1 0 1 20 1 0 300 0 1 4000 0 0 50000 1 0 1 2 2 3 2 4 1 5 Output 4 Input 5 10000 0 1 2000 1 0 300 0 1 40 0 0 1 1 0 1 2 2 3 2 4 1 5 Output 24000 Input 2 109 0 1 205 0 1 1 2 Output -1 -----Note----- The tree corresponding to samples $1$ and $2$ are: [Image] In sample $1$, we can choose node $1$ and $k = 4$ for a cost of $4 \cdot 1$ = $4$ and select nodes ${1, 2, 3, 5}$, shuffle their digits and get the desired digits in every node. In sample $2$, we can choose node $1$ and $k = 2$ for a cost of $10000 \cdot 2$, select nodes ${1, 5}$ and exchange their digits, and similarly, choose node $2$ and $k = 2$ for a cost of $2000 \cdot 2$, select nodes ${2, 3}$ and exchange their digits to get the desired digits in every node. In sample $3$, it is impossible to get the desired digits, because there is no node with digit $1$ initially. Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. The first algorithm for detecting a face on the image working in realtime was developed by Paul Viola and Michael Jones in 2001. A part of the algorithm is a procedure that computes Haar features. As part of this task, we consider a simplified model of this concept. Let's consider a rectangular image that is represented with a table of size n × m. The table elements are integers that specify the brightness of each pixel in the image. A feature also is a rectangular table of size n × m. Each cell of a feature is painted black or white. To calculate the value of the given feature at the given image, you must perform the following steps. First the table of the feature is put over the table of the image (without rotations or reflections), thus each pixel is entirely covered with either black or white cell. The value of a feature in the image is the value of W - B, where W is the total brightness of the pixels in the image, covered with white feature cells, and B is the total brightness of the pixels covered with black feature cells. Some examples of the most popular Haar features are given below. [Image] Your task is to determine the number of operations that are required to calculate the feature by using the so-called prefix rectangles. A prefix rectangle is any rectangle on the image, the upper left corner of which coincides with the upper left corner of the image. You have a variable value, whose value is initially zero. In one operation you can count the sum of pixel values at any prefix rectangle, multiply it by any integer and add to variable value. You are given a feature. It is necessary to calculate the minimum number of operations required to calculate the values of this attribute at an arbitrary image. For a better understanding of the statement, read the explanation of the first sample. -----Input----- The first line contains two space-separated integers n and m (1 ≤ n, m ≤ 100) — the number of rows and columns in the feature. Next n lines contain the description of the feature. Each line consists of m characters, the j-th character of the i-th line equals to "W", if this element of the feature is white and "B" if it is black. -----Output----- Print a single number — the minimum number of operations that you need to make to calculate the value of the feature. -----Examples----- Input 6 8 BBBBBBBB BBBBBBBB BBBBBBBB WWWWWWWW WWWWWWWW WWWWWWWW Output 2 Input 3 3 WBW BWW WWW Output 4 Input 3 6 WWBBWW WWBBWW WWBBWW Output 3 Input 4 4 BBBB BBBB BBBB BBBW Output 4 -----Note----- The first sample corresponds to feature B, the one shown in the picture. The value of this feature in an image of size 6 × 8 equals to the difference of the total brightness of the pixels in the lower and upper half of the image. To calculate its value, perform the following two operations: add the sum of pixels in the prefix rectangle with the lower right corner in the 6-th row and 8-th column with coefficient 1 to the variable value (the rectangle is indicated by a red frame); $\left. \begin{array}{\|r\|r\|r\|r\|r\|r\|r\|r\|} \hline 1 & {1} & {1} & {1} & {1} & {1} & {1} & {1} \\ \hline 1 & {1} & {1} & {1} & {1} & {1} & {1} & {1} \\ \hline 1 & {1} & {1} & {1} & {1} & {1} & {1} & {1} \\ \hline 1 & {1} & {1} & {1} & {1} & {1} & {1} & {1} \\ \hline 1 & {1} & {1} & {1} & {1} & {1} & {1} & {1} \\ \hline 1 & {1} & {1} & {1} & {1} & {1} & {1} & {1} \\ \hline \end{array} \right.$ add the number of pixels in the prefix rectangle with the lower right corner in the 3-rd row and 8-th column with coefficient - 2 and variable value. [Image] Thus, all the pixels in the lower three rows of the image will be included with factor 1, and all pixels in the upper three rows of the image will be included with factor 1 - 2 = - 1, as required. Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. The end of the school year is near and Ms. Manana, the teacher, will soon have to say goodbye to a yet another class. She decided to prepare a goodbye present for her n students and give each of them a jigsaw puzzle (which, as wikipedia states, is a tiling puzzle that requires the assembly of numerous small, often oddly shaped, interlocking and tessellating pieces). The shop assistant told the teacher that there are m puzzles in the shop, but they might differ in difficulty and size. Specifically, the first jigsaw puzzle consists of f_1 pieces, the second one consists of f_2 pieces and so on. Ms. Manana doesn't want to upset the children, so she decided that the difference between the numbers of pieces in her presents must be as small as possible. Let A be the number of pieces in the largest puzzle that the teacher buys and B be the number of pieces in the smallest such puzzle. She wants to choose such n puzzles that A - B is minimum possible. Help the teacher and find the least possible value of A - B. -----Input----- The first line contains space-separated integers n and m (2 ≤ n ≤ m ≤ 50). The second line contains m space-separated integers f_1, f_2, ..., f_{m} (4 ≤ f_{i} ≤ 1000) — the quantities of pieces in the puzzles sold in the shop. -----Output----- Print a single integer — the least possible difference the teacher can obtain. -----Examples----- Input 4 6 10 12 10 7 5 22 Output 5 -----Note----- Sample 1. The class has 4 students. The shop sells 6 puzzles. If Ms. Manana buys the first four puzzles consisting of 10, 12, 10 and 7 pieces correspondingly, then the difference between the sizes of the largest and the smallest puzzle will be equal to 5. It is impossible to obtain a smaller difference. Note that the teacher can also buy puzzles 1, 3, 4 and 5 to obtain the difference 5. Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. A group of n schoolboys decided to ride bikes. As nobody of them has a bike, the boys need to rent them. The renting site offered them m bikes. The renting price is different for different bikes, renting the j-th bike costs p_{j} rubles. In total, the boys' shared budget is a rubles. Besides, each of them has his own personal money, the i-th boy has b_{i} personal rubles. The shared budget can be spent on any schoolchildren arbitrarily, but each boy's personal money can be spent on renting only this boy's bike. Each boy can rent at most one bike, one cannot give his bike to somebody else. What maximum number of schoolboys will be able to ride bikes? What minimum sum of personal money will they have to spend in total to let as many schoolchildren ride bikes as possible? -----Input----- The first line of the input contains three integers n, m and a (1 ≤ n, m ≤ 10^5; 0 ≤ a ≤ 10^9). The second line contains the sequence of integers b_1, b_2, ..., b_{n} (1 ≤ b_{i} ≤ 10^4), where b_{i} is the amount of the i-th boy's personal money. The third line contains the sequence of integers p_1, p_2, ..., p_{m} (1 ≤ p_{j} ≤ 10^9), where p_{j} is the price for renting the j-th bike. -----Output----- Print two integers r and s, where r is the maximum number of schoolboys that can rent a bike and s is the minimum total personal money needed to rent r bikes. If the schoolchildren cannot rent any bikes, then r = s = 0. -----Examples----- Input 2 2 10 5 5 7 6 Output 2 3 Input 4 5 2 8 1 1 2 6 3 7 5 2 Output 3 8 -----Note----- In the first sample both schoolchildren can rent a bike. For instance, they can split the shared budget in half (5 rubles each). In this case one of them will have to pay 1 ruble from the personal money and the other one will have to pay 2 rubles from the personal money. In total, they spend 3 rubles of their personal money. This way of distribution of money minimizes the amount of spent personal money. Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. Let's call a number k-good if it contains all digits not exceeding k (0, ..., k). You've got a number k and an array a containing n numbers. Find out how many k-good numbers are in a (count each number every time it occurs in array a). -----Input----- The first line contains integers n and k (1 ≤ n ≤ 100, 0 ≤ k ≤ 9). The i-th of the following n lines contains integer a_{i} without leading zeroes (1 ≤ a_{i} ≤ 10^9). -----Output----- Print a single integer — the number of k-good numbers in a. -----Examples----- Input 10 6 1234560 1234560 1234560 1234560 1234560 1234560 1234560 1234560 1234560 1234560 Output 10 Input 2 1 1 10 Output 1 Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. Little Tanya decided to present her dad a postcard on his Birthday. She has already created a message — string s of length n, consisting of uppercase and lowercase English letters. Tanya can't write yet, so she found a newspaper and decided to cut out the letters and glue them into the postcard to achieve string s. The newspaper contains string t, consisting of uppercase and lowercase English letters. We know that the length of string t greater or equal to the length of the string s. The newspaper may possibly have too few of some letters needed to make the text and too many of some other letters. That's why Tanya wants to cut some n letters out of the newspaper and make a message of length exactly n, so that it looked as much as possible like s. If the letter in some position has correct value and correct letter case (in the string s and in the string that Tanya will make), then she shouts joyfully "YAY!", and if the letter in the given position has only the correct value but it is in the wrong case, then the girl says "WHOOPS". Tanya wants to make such message that lets her shout "YAY!" as much as possible. If there are multiple ways to do this, then her second priority is to maximize the number of times she says "WHOOPS". Your task is to help Tanya make the message. -----Input----- The first line contains line s (1 ≤ \|s\| ≤ 2·10^5), consisting of uppercase and lowercase English letters — the text of Tanya's message. The second line contains line t (\|s\| ≤ \|t\| ≤ 2·10^5), consisting of uppercase and lowercase English letters — the text written in the newspaper. Here \|a\| means the length of the string a. -----Output----- Print two integers separated by a space: the first number is the number of times Tanya shouts "YAY!" while making the message, the second number is the number of times Tanya says "WHOOPS" while making the message. -----Examples----- Input AbC DCbA Output 3 0 Input ABC abc Output 0 3 Input abacaba AbaCaBA Output 3 4 Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. Iahub is training for the IOI. What is a better way to train than playing a Zuma-like game? There are n balls put in a row. Each ball is colored in one of k colors. Initially the row doesn't contain three or more contiguous balls with the same color. Iahub has a single ball of color x. He can insert his ball at any position in the row (probably, between two other balls). If at any moment there are three or more contiguous balls of the same color in the row, they are destroyed immediately. This rule is applied multiple times, until there are no more sets of 3 or more contiguous balls of the same color. For example, if Iahub has the row of balls [black, black, white, white, black, black] and a white ball, he can insert the ball between two white balls. Thus three white balls are destroyed, and then four black balls become contiguous, so all four balls are destroyed. The row will not contain any ball in the end, so Iahub can destroy all 6 balls. Iahub wants to destroy as many balls as possible. You are given the description of the row of balls, and the color of Iahub's ball. Help Iahub train for the IOI by telling him the maximum number of balls from the row he can destroy. -----Input----- The first line of input contains three integers: n (1 ≤ n ≤ 100), k (1 ≤ k ≤ 100) and x (1 ≤ x ≤ k). The next line contains n space-separated integers c_1, c_2, ..., c_{n} (1 ≤ c_{i} ≤ k). Number c_{i} means that the i-th ball in the row has color c_{i}. It is guaranteed that the initial row of balls will never contain three or more contiguous balls of the same color. -----Output----- Print a single integer — the maximum number of balls Iahub can destroy. -----Examples----- Input 6 2 2 1 1 2 2 1 1 Output 6 Input 1 1 1 1 Output 0 Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. A country has n cities. Initially, there is no road in the country. One day, the king decides to construct some roads connecting pairs of cities. Roads can be traversed either way. He wants those roads to be constructed in such a way that it is possible to go from each city to any other city by traversing at most two roads. You are also given m pairs of cities — roads cannot be constructed between these pairs of cities. Your task is to construct the minimum number of roads that still satisfy the above conditions. The constraints will guarantee that this is always possible. -----Input----- The first line consists of two integers n and m $(1 \leq n \leq 10^{3}, 0 \leq m < \frac{n}{2})$. Then m lines follow, each consisting of two integers a_{i} and b_{i} (1 ≤ a_{i}, b_{i} ≤ n, a_{i} ≠ b_{i}), which means that it is not possible to construct a road connecting cities a_{i} and b_{i}. Consider the cities are numbered from 1 to n. It is guaranteed that every pair of cities will appear at most once in the input. -----Output----- You should print an integer s: the minimum number of roads that should be constructed, in the first line. Then s lines should follow, each consisting of two integers a_{i} and b_{i} (1 ≤ a_{i}, b_{i} ≤ n, a_{i} ≠ b_{i}), which means that a road should be constructed between cities a_{i} and b_{i}. If there are several solutions, you may print any of them. -----Examples----- Input 4 1 1 3 Output 3 1 2 4 2 2 3 -----Note----- This is one possible solution of the example: [Image] These are examples of wrong solutions: [Image] The above solution is wrong because it doesn't use the minimum number of edges (4 vs 3). In addition, it also tries to construct a road between cities 1 and 3, while the input specifies that it is not allowed to construct a road between the pair. [Image] The above solution is wrong because you need to traverse at least 3 roads to go from city 1 to city 3, whereas in your country it must be possible to go from any city to another by traversing at most 2 roads. [Image] Finally, the above solution is wrong because it must be possible to go from any city to another, whereas it is not possible in this country to go from city 1 to 3, 2 to 3, and 4 to 3. Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. One day Vasya was on a physics practical, performing the task on measuring the capacitance. He followed the teacher's advice and did as much as n measurements, and recorded the results in the notebook. After that he was about to show the results to the teacher, but he remembered that at the last lesson, the teacher had made his friend Petya redo the experiment because the largest and the smallest results differed by more than two times. Vasya is lazy, and he does not want to redo the experiment. He wants to do the task and go home play computer games. So he decided to cheat: before Vasya shows the measurements to the teacher, he will erase some of them, so as to make the largest and the smallest results of the remaining measurements differ in no more than two times. In other words, if the remaining measurements have the smallest result x, and the largest result y, then the inequality y ≤ 2·x must fulfill. Of course, to avoid the teacher's suspicion, Vasya wants to remove as few measurement results as possible from his notes. Help Vasya, find what minimum number of measurement results he will have to erase from his notes so that the largest and the smallest of the remaining results of the measurements differed in no more than two times. -----Input----- The first line contains integer n (2 ≤ n ≤ 10^5) — the number of measurements Vasya made. The second line contains n integers c_1, c_2, ..., c_{n} (1 ≤ c_{i} ≤ 5000) — the results of the measurements. The numbers on the second line are separated by single spaces. -----Output----- Print a single integer — the minimum number of results Vasya will have to remove. -----Examples----- Input 6 4 5 3 8 3 7 Output 2 Input 4 4 3 2 4 Output 0 -----Note----- In the first sample you can remove the fourth and the sixth measurement results (values 8 and 7). Then the maximum of the remaining values will be 5, and the minimum one will be 3. Or else, you can remove the third and fifth results (both equal 3). After that the largest remaining result will be 8, and the smallest one will be 4. Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. Today Adilbek is taking his probability theory test. Unfortunately, when Adilbek arrived at the university, there had already been a long queue of students wanting to take the same test. Adilbek has estimated that he will be able to start the test only $T$ seconds after coming. Fortunately, Adilbek can spend time without revising any boring theorems or formulas. He has an app on this smartphone which contains $n$ Japanese crosswords to solve. Adilbek has decided to solve them all one by one in the order they are listed in the app, without skipping any crossword. For each crossword, a number $t_i$ is given that represents the time it takes an average crossword expert to solve this crossword (the time is given in seconds). Adilbek is a true crossword expert, but, unfortunately, he is sometimes unlucky in choosing the way to solve the crossword. So, it takes him either $t_i$ seconds or $t_i + 1$ seconds to solve the $i$-th crossword, equiprobably (with probability $\frac{1}{2}$ he solves the crossword in exactly $t_i$ seconds, and with probability $\frac{1}{2}$ he has to spend an additional second to finish the crossword). All these events are independent. After $T$ seconds pass (or after solving the last crossword, if he manages to do it in less than $T$ seconds), Adilbek closes the app (if he finishes some crossword at the same moment, that crossword is considered solved; otherwise Adilbek does not finish solving the current crossword at all). He thinks it would be an interesting probability theory problem to calculate $E$ — the expected number of crosswords he will be able to solve completely. Can you calculate it? Recall that the expected value of a discrete random variable is the probability-weighted average of all possible values — in this problem it means that the expected value of the number of solved crosswords can be calculated as $E = \sum \limits_{i = 0}^{n} i p_i$, where $p_i$ is the probability that Adilbek will solve exactly $i$ crosswords. We can represent $E$ as rational fraction $\frac{P}{Q}$ with $Q > 0$. To give the answer, you should print $P \cdot Q^{-1} \bmod (10^9 + 7)$. -----Input----- The first line contains two integers $n$ and $T$ ($1 \le n \le 2 \cdot 10^5$, $1 \le T \le 2 \cdot 10^{14}$) — the number of crosswords and the time Adilbek has to spend, respectively. The second line contains $n$ integers $t_1, t_2, \dots, t_n$ ($1 \le t_i \le 10^9$), where $t_i$ is the time it takes a crossword expert to solve the $i$-th crossword. Note that Adilbek solves the crosswords in the order they are given in the input without skipping any of them. -----Output----- Print one integer — the expected value of the number of crosswords Adilbek solves in $T$ seconds, expressed in the form of $P \cdot Q^{-1} \bmod (10^9 + 7)$. -----Examples----- Input 3 5 2 2 2 Output 750000007 Input 3 5 2 1 2 Output 125000003 -----Note----- The answer for the first sample is equal to $\frac{14}{8}$. The answer for the second sample is equal to $\frac{17}{8}$. Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. Alyona decided to go on a diet and went to the forest to get some apples. There she unexpectedly found a magic rooted tree with root in the vertex 1, every vertex and every edge of which has a number written on. The girl noticed that some of the tree's vertices are sad, so she decided to play with them. Let's call vertex v sad if there is a vertex u in subtree of vertex v such that dist(v, u) > a_{u}, where a_{u} is the number written on vertex u, dist(v, u) is the sum of the numbers written on the edges on the path from v to u. Leaves of a tree are vertices connected to a single vertex by a single edge, but the root of a tree is a leaf if and only if the tree consists of a single vertex — root. Thus Alyona decided to remove some of tree leaves until there will be no any sad vertex left in the tree. What is the minimum number of leaves Alyona needs to remove? -----Input----- In the first line of the input integer n (1 ≤ n ≤ 10^5) is given — the number of vertices in the tree. In the second line the sequence of n integers a_1, a_2, ..., a_{n} (1 ≤ a_{i} ≤ 10^9) is given, where a_{i} is the number written on vertex i. The next n - 1 lines describe tree edges: i^{th} of them consists of two integers p_{i} and c_{i} (1 ≤ p_{i} ≤ n, - 10^9 ≤ c_{i} ≤ 10^9), meaning that there is an edge connecting vertices i + 1 and p_{i} with number c_{i} written on it. -----Output----- Print the only integer — the minimum number of leaves Alyona needs to remove such that there will be no any sad vertex left in the tree. -----Example----- Input 9 88 22 83 14 95 91 98 53 11 3 24 7 -8 1 67 1 64 9 65 5 12 6 -80 3 8 Output 5 -----Note----- The following image represents possible process of removing leaves from the tree: [Image] Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. Yaroslav thinks that two strings s and w, consisting of digits and having length n are non-comparable if there are two numbers, i and j (1 ≤ i, j ≤ n), such that s_{i} > w_{i} and s_{j} < w_{j}. Here sign s_{i} represents the i-th digit of string s, similarly, w_{j} represents the j-th digit of string w. A string's template is a string that consists of digits and question marks ("?"). Yaroslav has two string templates, each of them has length n. Yaroslav wants to count the number of ways to replace all question marks by some integers in both templates, so as to make the resulting strings incomparable. Note that the obtained strings can contain leading zeroes and that distinct question marks can be replaced by distinct or the same integers. Help Yaroslav, calculate the remainder after dividing the described number of ways by 1000000007 (10^9 + 7). -----Input----- The first line contains integer n (1 ≤ n ≤ 10^5) — the length of both templates. The second line contains the first template — a string that consists of digits and characters "?". The string's length equals n. The third line contains the second template in the same format. -----Output----- In a single line print the remainder after dividing the answer to the problem by number 1000000007 (10^9 + 7). -----Examples----- Input 2 90 09 Output 1 Input 2 11 55 Output 0 Input 5 ????? ????? Output 993531194 -----Note----- The first test contains no question marks and both strings are incomparable, so the answer is 1. The second test has no question marks, but the given strings are comparable, so the answer is 0. Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. You have a Petri dish with bacteria and you are preparing to dive into the harsh micro-world. But, unfortunately, you don't have any microscope nearby, so you can't watch them. You know that you have $n$ bacteria in the Petri dish and size of the $i$-th bacteria is $a_i$. Also you know intergalactic positive integer constant $K$. The $i$-th bacteria can swallow the $j$-th bacteria if and only if $a_i > a_j$ and $a_i \le a_j + K$. The $j$-th bacteria disappear, but the $i$-th bacteria doesn't change its size. The bacteria can perform multiple swallows. On each swallow operation any bacteria $i$ can swallow any bacteria $j$ if $a_i > a_j$ and $a_i \le a_j + K$. The swallow operations go one after another. For example, the sequence of bacteria sizes $a=[101, 53, 42, 102, 101, 55, 54]$ and $K=1$. The one of possible sequences of swallows is: $[101, 53, 42, 102, \underline{101}, 55, 54]$ $\to$ $[101, \underline{53}, 42, 102, 55, 54]$ $\to$ $[\underline{101}, 42, 102, 55, 54]$ $\to$ $[42, 102, 55, \underline{54}]$ $\to$ $[42, 102, 55]$. In total there are $3$ bacteria remained in the Petri dish. Since you don't have a microscope, you can only guess, what the minimal possible number of bacteria can remain in your Petri dish when you finally will find any microscope. -----Input----- The first line contains two space separated positive integers $n$ and $K$ ($1 \le n \le 2 \cdot 10^5$, $1 \le K \le 10^6$) — number of bacteria and intergalactic constant $K$. The second line contains $n$ space separated integers $a_1, a_2, \dots, a_n$ ($1 \le a_i \le 10^6$) — sizes of bacteria you have. -----Output----- Print the only integer — minimal possible number of bacteria can remain. -----Examples----- Input 7 1 101 53 42 102 101 55 54 Output 3 Input 6 5 20 15 10 15 20 25 Output 1 Input 7 1000000 1 1 1 1 1 1 1 Output 7 -----Note----- The first example is clarified in the problem statement. In the second example an optimal possible sequence of swallows is: $[20, 15, 10, 15, \underline{20}, 25]$ $\to$ $[20, 15, 10, \underline{15}, 25]$ $\to$ $[20, 15, \underline{10}, 25]$ $\to$ $[20, \underline{15}, 25]$ $\to$ $[\underline{20}, 25]$ $\to$ $[25]$. In the third example no bacteria can swallow any other bacteria. Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. Ivan unexpectedly saw a present from one of his previous birthdays. It is array of $n$ numbers from $1$ to $200$. Array is old and some numbers are hard to read. Ivan remembers that for all elements at least one of its neighbours ls not less than it, more formally: $a_{1} \le a_{2}$, $a_{n} \le a_{n-1}$ and $a_{i} \le max(a_{i-1}, \,\, a_{i+1})$ for all $i$ from $2$ to $n-1$. Ivan does not remember the array and asks to find the number of ways to restore it. Restored elements also should be integers from $1$ to $200$. Since the number of ways can be big, print it modulo $998244353$. -----Input----- First line of input contains one integer $n$ ($2 \le n \le 10^{5}$) — size of the array. Second line of input contains $n$ integers $a_{i}$ — elements of array. Either $a_{i} = -1$ or $1 \le a_{i} \le 200$. $a_{i} = -1$ means that $i$-th element can't be read. -----Output----- Print number of ways to restore the array modulo $998244353$. -----Examples----- Input 3 1 -1 2 Output 1 Input 2 -1 -1 Output 200 -----Note----- In the first example, only possible value of $a_{2}$ is $2$. In the second example, $a_{1} = a_{2}$ so there are $200$ different values because all restored elements should be integers between $1$ and $200$. Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. Recently Pashmak has been employed in a transportation company. The company has k buses and has a contract with a school which has n students. The school planned to take the students to d different places for d days (each day in one place). Each day the company provides all the buses for the trip. Pashmak has to arrange the students in the buses. He wants to arrange the students in a way that no two students become close friends. In his ridiculous idea, two students will become close friends if and only if they are in the same buses for all d days. Please help Pashmak with his weird idea. Assume that each bus has an unlimited capacity. -----Input----- The first line of input contains three space-separated integers n, k, d (1 ≤ n, d ≤ 1000; 1 ≤ k ≤ 10^9). -----Output----- If there is no valid arrangement just print -1. Otherwise print d lines, in each of them print n integers. The j-th integer of the i-th line shows which bus the j-th student has to take on the i-th day. You can assume that the buses are numbered from 1 to k. -----Examples----- Input 3 2 2 Output 1 1 2 1 2 1 Input 3 2 1 Output -1 -----Note----- Note that two students become close friends only if they share a bus each day. But the bus they share can differ from day to day. Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. You've got an n × m matrix. The matrix consists of integers. In one move, you can apply a single transformation to the matrix: choose an arbitrary element of the matrix and increase it by 1. Each element can be increased an arbitrary number of times. You are really curious about prime numbers. Let us remind you that a prime number is a positive integer that has exactly two distinct positive integer divisors: itself and number one. For example, numbers 2, 3, 5 are prime and numbers 1, 4, 6 are not. A matrix is prime if at least one of the two following conditions fulfills: the matrix has a row with prime numbers only; the matrix has a column with prime numbers only; Your task is to count the minimum number of moves needed to get a prime matrix from the one you've got. -----Input----- The first line contains two integers n, m (1 ≤ n, m ≤ 500) — the number of rows and columns in the matrix, correspondingly. Each of the following n lines contains m integers — the initial matrix. All matrix elements are positive integers. All numbers in the initial matrix do not exceed 10^5. The numbers in the lines are separated by single spaces. -----Output----- Print a single integer — the minimum number of moves needed to get a prime matrix from the one you've got. If you've got a prime matrix, print 0. -----Examples----- Input 3 3 1 2 3 5 6 1 4 4 1 Output 1 Input 2 3 4 8 8 9 2 9 Output 3 Input 2 2 1 3 4 2 Output 0 -----Note----- In the first sample you need to increase number 1 in cell (1, 1). Thus, the first row will consist of prime numbers: 2, 2, 3. In the second sample you need to increase number 8 in cell (1, 2) three times. Thus, the second column will consist of prime numbers: 11, 2. In the third sample you don't have to do anything as the second column already consists of prime numbers: 3, 2. Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. Shaass has n books. He wants to make a bookshelf for all his books. He wants the bookshelf's dimensions to be as small as possible. The thickness of the i-th book is t_{i} and its pages' width is equal to w_{i}. The thickness of each book is either 1 or 2. All books have the same page heights. $1$ Shaass puts the books on the bookshelf in the following way. First he selects some of the books and put them vertically. Then he puts the rest of the books horizontally above the vertical books. The sum of the widths of the horizontal books must be no more than the total thickness of the vertical books. A sample arrangement of the books is depicted in the figure. [Image] Help Shaass to find the minimum total thickness of the vertical books that we can achieve. -----Input----- The first line of the input contains an integer n, (1 ≤ n ≤ 100). Each of the next n lines contains two integers t_{i} and w_{i} denoting the thickness and width of the i-th book correspondingly, (1 ≤ t_{i} ≤ 2, 1 ≤ w_{i} ≤ 100). -----Output----- On the only line of the output print the minimum total thickness of the vertical books that we can achieve. -----Examples----- Input 5 1 12 1 3 2 15 2 5 2 1 Output 5 Input 3 1 10 2 1 2 4 Output 3 Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. The Saratov State University Olympiad Programmers Training Center (SSU OPTC) has n students. For each student you know the number of times he/she has participated in the ACM ICPC world programming championship. According to the ACM ICPC rules, each person can participate in the world championship at most 5 times. The head of the SSU OPTC is recently gathering teams to participate in the world championship. Each team must consist of exactly three people, at that, any person cannot be a member of two or more teams. What maximum number of teams can the head make if he wants each team to participate in the world championship with the same members at least k times? -----Input----- The first line contains two integers, n and k (1 ≤ n ≤ 2000; 1 ≤ k ≤ 5). The next line contains n integers: y_1, y_2, ..., y_{n} (0 ≤ y_{i} ≤ 5), where y_{i} shows the number of times the i-th person participated in the ACM ICPC world championship. -----Output----- Print a single number — the answer to the problem. -----Examples----- Input 5 2 0 4 5 1 0 Output 1 Input 6 4 0 1 2 3 4 5 Output 0 Input 6 5 0 0 0 0 0 0 Output 2 -----Note----- In the first sample only one team could be made: the first, the fourth and the fifth participants. In the second sample no teams could be created. In the third sample two teams could be created. Any partition into two teams fits. Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. You are given a tree consisting of $n$ vertices. A tree is an undirected connected acyclic graph. [Image] Example of a tree. You have to paint each vertex into one of three colors. For each vertex, you know the cost of painting it in every color. You have to paint the vertices so that any path consisting of exactly three distinct vertices does not contain any vertices with equal colors. In other words, let's consider all triples $(x, y, z)$ such that $x \neq y, y \neq z, x \neq z$, $x$ is connected by an edge with $y$, and $y$ is connected by an edge with $z$. The colours of $x$, $y$ and $z$ should be pairwise distinct. Let's call a painting which meets this condition good. You have to calculate the minimum cost of a good painting and find one of the optimal paintings. If there is no good painting, report about it. -----Input----- The first line contains one integer $n$ $(3 \le n \le 100\,000)$ — the number of vertices. The second line contains a sequence of integers $c_{1, 1}, c_{1, 2}, \dots, c_{1, n}$ $(1 \le c_{1, i} \le 10^{9})$, where $c_{1, i}$ is the cost of painting the $i$-th vertex into the first color. The third line contains a sequence of integers $c_{2, 1}, c_{2, 2}, \dots, c_{2, n}$ $(1 \le c_{2, i} \le 10^{9})$, where $c_{2, i}$ is the cost of painting the $i$-th vertex into the second color. The fourth line contains a sequence of integers $c_{3, 1}, c_{3, 2}, \dots, c_{3, n}$ $(1 \le c_{3, i} \le 10^{9})$, where $c_{3, i}$ is the cost of painting the $i$-th vertex into the third color. Then $(n - 1)$ lines follow, each containing two integers $u_j$ and $v_j$ $(1 \le u_j, v_j \le n, u_j \neq v_j)$ — the numbers of vertices connected by the $j$-th undirected edge. It is guaranteed that these edges denote a tree. -----Output----- If there is no good painting, print $-1$. Otherwise, print the minimum cost of a good painting in the first line. In the second line print $n$ integers $b_1, b_2, \dots, b_n$ $(1 \le b_i \le 3)$, where the $i$-th integer should denote the color of the $i$-th vertex. If there are multiple good paintings with minimum cost, print any of them. -----Examples----- Input 3 3 2 3 4 3 2 3 1 3 1 2 2 3 Output 6 1 3 2 Input 5 3 4 2 1 2 4 2 1 5 4 5 3 2 1 1 1 2 3 2 4 3 5 3 Output -1 Input 5 3 4 2 1 2 4 2 1 5 4 5 3 2 1 1 1 2 3 2 4 3 5 4 Output 9 1 3 2 1 3 -----Note----- All vertices should be painted in different colors in the first example. The optimal way to do it is to paint the first vertex into color $1$, the second vertex — into color $3$, and the third vertex — into color $2$. The cost of this painting is $3 + 2 + 1 = 6$. Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. All our characters have hobbies. The same is true for Fedor. He enjoys shopping in the neighboring supermarket. The goods in the supermarket have unique integer ids. Also, for every integer there is a product with id equal to this integer. Fedor has n discount coupons, the i-th of them can be used with products with ids ranging from l_{i} to r_{i}, inclusive. Today Fedor wants to take exactly k coupons with him. Fedor wants to choose the k coupons in such a way that the number of such products x that all coupons can be used with this product x is as large as possible (for better understanding, see examples). Fedor wants to save his time as well, so he asks you to choose coupons for him. Help Fedor! -----Input----- The first line contains two integers n and k (1 ≤ k ≤ n ≤ 3·10^5) — the number of coupons Fedor has, and the number of coupons he wants to choose. Each of the next n lines contains two integers l_{i} and r_{i} ( - 10^9 ≤ l_{i} ≤ r_{i} ≤ 10^9) — the description of the i-th coupon. The coupons can be equal. -----Output----- In the first line print single integer — the maximum number of products with which all the chosen coupons can be used. The products with which at least one coupon cannot be used shouldn't be counted. In the second line print k distinct integers p_1, p_2, ..., p_{k} (1 ≤ p_{i} ≤ n) — the ids of the coupons which Fedor should choose. If there are multiple answers, print any of them. -----Examples----- Input 4 2 1 100 40 70 120 130 125 180 Output 31 1 2 Input 3 2 1 12 15 20 25 30 Output 0 1 2 Input 5 2 1 10 5 15 14 50 30 70 99 100 Output 21 3 4 -----Note----- In the first example if we take the first two coupons then all the products with ids in range [40, 70] can be bought with both coupons. There are 31 products in total. In the second example, no product can be bought with two coupons, that is why the answer is 0. Fedor can choose any two coupons in this example. Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. You are given a string $s=s_1s_2\dots s_n$ of length $n$, which only contains digits $1$, $2$, ..., $9$. A substring $s[l \dots r]$ of $s$ is a string $s_l s_{l + 1} s_{l + 2} \ldots s_r$. A substring $s[l \dots r]$ of $s$ is called even if the number represented by it is even. Find the number of even substrings of $s$. Note, that even if some substrings are equal as strings, but have different $l$ and $r$, they are counted as different substrings. -----Input----- The first line contains an integer $n$ ($1 \le n \le 65000$) — the length of the string $s$. The second line contains a string $s$ of length $n$. The string $s$ consists only of digits $1$, $2$, ..., $9$. -----Output----- Print the number of even substrings of $s$. -----Examples----- Input 4 1234 Output 6 Input 4 2244 Output 10 -----Note----- In the first example, the $[l, r]$ pairs corresponding to even substrings are: $s[1 \dots 2]$ $s[2 \dots 2]$ $s[1 \dots 4]$ $s[2 \dots 4]$ $s[3 \dots 4]$ $s[4 \dots 4]$ In the second example, all $10$ substrings of $s$ are even substrings. Note, that while substrings $s[1 \dots 1]$ and $s[2 \dots 2]$ both define the substring "2", they are still counted as different substrings. Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. Inna and Dima bought a table of size n × m in the shop. Each cell of the table contains a single letter: "D", "I", "M", "A". Inna loves Dima, so she wants to go through his name as many times as possible as she moves through the table. For that, Inna acts as follows: initially, Inna chooses some cell of the table where letter "D" is written; then Inna can move to some side-adjacent table cell that contains letter "I"; then from this cell she can go to one of the side-adjacent table cells that contains the written letter "M"; then she can go to a side-adjacent cell that contains letter "A". Then Inna assumes that she has gone through her sweetheart's name; Inna's next move can be going to one of the side-adjacent table cells that contains letter "D" and then walk on through name DIMA in the similar manner. Inna never skips a letter. So, from the letter "D" she always goes to the letter "I", from the letter "I" she always goes the to letter "M", from the letter "M" she always goes to the letter "A", and from the letter "A" she always goes to the letter "D". Depending on the choice of the initial table cell, Inna can go through name DIMA either an infinite number of times or some positive finite number of times or she can't go through his name once. Help Inna find out what maximum number of times she can go through name DIMA. -----Input----- The first line of the input contains two integers n and m (1 ≤ n, m ≤ 10^3). Then follow n lines that describe Inna and Dima's table. Each line contains m characters. Each character is one of the following four characters: "D", "I", "M", "A". Note that it is not guaranteed that the table contains at least one letter "D". -----Output----- If Inna cannot go through name DIMA once, print on a single line "Poor Dima!" without the quotes. If there is the infinite number of names DIMA Inna can go through, print "Poor Inna!" without the quotes. Otherwise print a single integer — the maximum number of times Inna can go through name DIMA. -----Examples----- Input 1 2 DI Output Poor Dima! Input 2 2 MA ID Output Poor Inna! Input 5 5 DIMAD DIMAI DIMAM DDMAA AAMID Output 4 -----Note----- Notes to the samples: In the first test sample, Inna cannot go through name DIMA a single time. In the second test sample, Inna can go through the infinite number of words DIMA. For that, she should move in the clockwise direction starting from the lower right corner. In the third test sample the best strategy is to start from the cell in the upper left corner of the table. Starting from this cell, Inna can go through name DIMA four times. Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. The Cybernetics Failures (CF) organisation made a prototype of a bomb technician robot. To find the possible problems it was decided to carry out a series of tests. At the beginning of each test the robot prototype will be placed in cell (x_0, y_0) of a rectangular squared field of size x × y, after that a mine will be installed into one of the squares of the field. It is supposed to conduct exactly x·y tests, each time a mine is installed into a square that has never been used before. The starting cell of the robot always remains the same. After placing the objects on the field the robot will have to run a sequence of commands given by string s, consisting only of characters 'L', 'R', 'U', 'D'. These commands tell the robot to move one square to the left, to the right, up or down, or stay idle if moving in the given direction is impossible. As soon as the robot fulfills all the sequence of commands, it will blow up due to a bug in the code. But if at some moment of time the robot is at the same square with the mine, it will also blow up, but not due to a bug in the code. Moving to the left decreases coordinate y, and moving to the right increases it. Similarly, moving up decreases the x coordinate, and moving down increases it. The tests can go on for very long, so your task is to predict their results. For each k from 0 to length(s) your task is to find in how many tests the robot will run exactly k commands before it blows up. -----Input----- The first line of the input contains four integers x, y, x_0, y_0 (1 ≤ x, y ≤ 500, 1 ≤ x_0 ≤ x, 1 ≤ y_0 ≤ y) — the sizes of the field and the starting coordinates of the robot. The coordinate axis X is directed downwards and axis Y is directed to the right. The second line contains a sequence of commands s, which should be fulfilled by the robot. It has length from 1 to 100 000 characters and only consists of characters 'L', 'R', 'U', 'D'. -----Output----- Print the sequence consisting of (length(s) + 1) numbers. On the k-th position, starting with zero, print the number of tests where the robot will run exactly k commands before it blows up. -----Examples----- Input 3 4 2 2 UURDRDRL Output 1 1 0 1 1 1 1 0 6 Input 2 2 2 2 ULD Output 1 1 1 1 -----Note----- In the first sample, if we exclude the probable impact of the mines, the robot's route will look like that: $(2,2) \rightarrow(1,2) \rightarrow(1,2) \rightarrow(1,3) \rightarrow(2,3) \rightarrow(2,4) \rightarrow(3,4) \rightarrow(3,4) \rightarrow(3,3)$. Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. Pasha decided to invite his friends to a tea party. For that occasion, he has a large teapot with the capacity of w milliliters and 2n tea cups, each cup is for one of Pasha's friends. The i-th cup can hold at most a_{i} milliliters of water. It turned out that among Pasha's friends there are exactly n boys and exactly n girls and all of them are going to come to the tea party. To please everyone, Pasha decided to pour the water for the tea as follows: Pasha can boil the teapot exactly once by pouring there at most w milliliters of water; Pasha pours the same amount of water to each girl; Pasha pours the same amount of water to each boy; if each girl gets x milliliters of water, then each boy gets 2x milliliters of water. In the other words, each boy should get two times more water than each girl does. Pasha is very kind and polite, so he wants to maximize the total amount of the water that he pours to his friends. Your task is to help him and determine the optimum distribution of cups between Pasha's friends. -----Input----- The first line of the input contains two integers, n and w (1 ≤ n ≤ 10^5, 1 ≤ w ≤ 10^9) — the number of Pasha's friends that are boys (equal to the number of Pasha's friends that are girls) and the capacity of Pasha's teapot in milliliters. The second line of the input contains the sequence of integers a_{i} (1 ≤ a_{i} ≤ 10^9, 1 ≤ i ≤ 2n) — the capacities of Pasha's tea cups in milliliters. -----Output----- Print a single real number — the maximum total amount of water in milliliters that Pasha can pour to his friends without violating the given conditions. Your answer will be considered correct if its absolute or relative error doesn't exceed 10^{ - 6}. -----Examples----- Input 2 4 1 1 1 1 Output 3 Input 3 18 4 4 4 2 2 2 Output 18 Input 1 5 2 3 Output 4.5 -----Note----- Pasha also has candies that he is going to give to girls but that is another task... Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. Let $n$ be an integer. Consider all permutations on integers $1$ to $n$ in lexicographic order, and concatenate them into one big sequence $P$. For example, if $n = 3$, then $P = [1, 2, 3, 1, 3, 2, 2, 1, 3, 2, 3, 1, 3, 1, 2, 3, 2, 1]$. The length of this sequence is $n \cdot n!$. Let $1 \leq i \leq j \leq n \cdot n!$ be a pair of indices. We call the sequence $(P_i, P_{i+1}, \dots, P_{j-1}, P_j)$ a subarray of $P$. You are given $n$. Find the number of distinct subarrays of $P$. Since this number may be large, output it modulo $998244353$ (a prime number). -----Input----- The only line contains one integer $n$ ($1 \leq n \leq 10^6$), as described in the problem statement. -----Output----- Output a single integer — the number of distinct subarrays, modulo $998244353$. -----Examples----- Input 2 Output 8 Input 10 Output 19210869 -----Note----- In the first example, the sequence $P = [1, 2, 2, 1]$. It has eight distinct subarrays: $[1]$, $[2]$, $[1, 2]$, $[2, 1]$, $[2, 2]$, $[1, 2, 2]$, $[2, 2, 1]$ and $[1, 2, 2, 1]$. Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. You're given an integer $n$. For every integer $i$ from $2$ to $n$, assign a positive integer $a_i$ such that the following conditions hold: For any pair of integers $(i,j)$, if $i$ and $j$ are coprime, $a_i \neq a_j$. The maximal value of all $a_i$ should be minimized (that is, as small as possible). A pair of integers is called coprime if their greatest common divisor is $1$. -----Input----- The only line contains the integer $n$ ($2 \le n \le 10^5$). -----Output----- Print $n-1$ integers, $a_2$, $a_3$, $\ldots$, $a_n$ ($1 \leq a_i \leq n$). If there are multiple solutions, print any of them. -----Examples----- Input 4 Output 1 2 1 Input 3 Output 2 1 -----Note----- In the first example, notice that $3$ and $4$ are coprime, so $a_3 \neq a_4$. Also, notice that $a=[1,2,3]$ satisfies the first condition, but it's not a correct answer because its maximal value is $3$. Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. The main city magazine offers its readers an opportunity to publish their ads. The format of the ad should be like this: There are space-separated non-empty words of lowercase and uppercase Latin letters. There are hyphen characters '-' in some words, their positions set word wrapping points. Word can include more than one hyphen. It is guaranteed that there are no adjacent spaces and no adjacent hyphens. No hyphen is adjacent to space. There are no spaces and no hyphens before the first word and after the last word. When the word is wrapped, the part of the word before hyphen and the hyphen itself stay on current line and the next part of the word is put on the next line. You can also put line break between two words, in that case the space stays on current line. Check notes for better understanding. The ad can occupy no more that k lines and should have minimal width. The width of the ad is the maximal length of string (letters, spaces and hyphens are counted) in it. You should write a program that will find minimal width of the ad. -----Input----- The first line contains number k (1 ≤ k ≤ 10^5). The second line contains the text of the ad — non-empty space-separated words of lowercase and uppercase Latin letters and hyphens. Total length of the ad don't exceed 10^6 characters. -----Output----- Output minimal width of the ad. -----Examples----- Input 4 garage for sa-le Output 7 Input 4 Edu-ca-tion-al Ro-unds are so fun Output 10 -----Note----- Here all spaces are replaced with dots. In the first example one of possible results after all word wraps looks like this: garage. for. sa- le The second example: Edu-ca- tion-al. Ro-unds. are.so.fun Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. Vanya walks late at night along a straight street of length l, lit by n lanterns. Consider the coordinate system with the beginning of the street corresponding to the point 0, and its end corresponding to the point l. Then the i-th lantern is at the point a_{i}. The lantern lights all points of the street that are at the distance of at most d from it, where d is some positive number, common for all lanterns. Vanya wonders: what is the minimum light radius d should the lanterns have to light the whole street? -----Input----- The first line contains two integers n, l (1 ≤ n ≤ 1000, 1 ≤ l ≤ 10^9) — the number of lanterns and the length of the street respectively. The next line contains n integers a_{i} (0 ≤ a_{i} ≤ l). Multiple lanterns can be located at the same point. The lanterns may be located at the ends of the street. -----Output----- Print the minimum light radius d, needed to light the whole street. The answer will be considered correct if its absolute or relative error doesn't exceed 10^{ - 9}. -----Examples----- Input 7 15 15 5 3 7 9 14 0 Output 2.5000000000 Input 2 5 2 5 Output 2.0000000000 -----Note----- Consider the second sample. At d = 2 the first lantern will light the segment [0, 4] of the street, and the second lantern will light segment [3, 5]. Thus, the whole street will be lit. Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. Generous sponsors of the olympiad in which Chloe and Vladik took part allowed all the participants to choose a prize for them on their own. Christmas is coming, so sponsors decided to decorate the Christmas tree with their prizes. They took n prizes for the contestants and wrote on each of them a unique id (integer from 1 to n). A gift i is characterized by integer a_{i} — pleasantness of the gift. The pleasantness of the gift can be positive, negative or zero. Sponsors placed the gift 1 on the top of the tree. All the other gifts hung on a rope tied to some other gift so that each gift hung on the first gift, possibly with a sequence of ropes and another gifts. Formally, the gifts formed a rooted tree with n vertices. The prize-giving procedure goes in the following way: the participants come to the tree one after another, choose any of the remaining gifts and cut the rope this prize hang on. Note that all the ropes which were used to hang other prizes on the chosen one are not cut. So the contestant gets the chosen gift as well as the all the gifts that hang on it, possibly with a sequence of ropes and another gifts. Our friends, Chloe and Vladik, shared the first place on the olympiad and they will choose prizes at the same time! To keep themselves from fighting, they decided to choose two different gifts so that the sets of the gifts that hang on them with a sequence of ropes and another gifts don't intersect. In other words, there shouldn't be any gift that hang both on the gift chosen by Chloe and on the gift chosen by Vladik. From all of the possible variants they will choose such pair of prizes that the sum of pleasantness of all the gifts that they will take after cutting the ropes is as large as possible. Print the maximum sum of pleasantness that Vladik and Chloe can get. If it is impossible for them to choose the gifts without fighting, print Impossible. -----Input----- The first line contains a single integer n (1 ≤ n ≤ 2·10^5) — the number of gifts. The next line contains n integers a_1, a_2, ..., a_{n} ( - 10^9 ≤ a_{i} ≤ 10^9) — the pleasantness of the gifts. The next (n - 1) lines contain two numbers each. The i-th of these lines contains integers u_{i} and v_{i} (1 ≤ u_{i}, v_{i} ≤ n, u_{i} ≠ v_{i}) — the description of the tree's edges. It means that gifts with numbers u_{i} and v_{i} are connected to each other with a rope. The gifts' ids in the description of the ropes can be given in arbirtary order: v_{i} hangs on u_{i} or u_{i} hangs on v_{i}. It is guaranteed that all the gifts hang on the first gift, possibly with a sequence of ropes and another gifts. -----Output----- If it is possible for Chloe and Vladik to choose prizes without fighting, print single integer — the maximum possible sum of pleasantness they can get together. Otherwise print Impossible. -----Examples----- Input 8 0 5 -1 4 3 2 6 5 1 2 2 4 2 5 1 3 3 6 6 7 6 8 Output 25 Input 4 1 -5 1 1 1 2 1 4 2 3 Output 2 Input 1 -1 Output Impossible Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. Xenia has a set of weights and pan scales. Each weight has an integer weight from 1 to 10 kilos. Xenia is going to play with scales and weights a little. For this, she puts weights on the scalepans, one by one. The first weight goes on the left scalepan, the second weight goes on the right scalepan, the third one goes on the left scalepan, the fourth one goes on the right scalepan and so on. Xenia wants to put the total of m weights on the scalepans. Simply putting weights on the scales is not interesting, so Xenia has set some rules. First, she does not put on the scales two consecutive weights of the same weight. That is, the weight that goes i-th should be different from the (i + 1)-th weight for any i (1 ≤ i < m). Second, every time Xenia puts a weight on some scalepan, she wants this scalepan to outweigh the other one. That is, the sum of the weights on the corresponding scalepan must be strictly greater than the sum on the other pan. You are given all types of weights available for Xenia. You can assume that the girl has an infinite number of weights of each specified type. Your task is to help Xenia lay m weights on the scales or to say that it can't be done. -----Input----- The first line contains a string consisting of exactly ten zeroes and ones: the i-th (i ≥ 1) character in the line equals "1" if Xenia has i kilo weights, otherwise the character equals "0". The second line contains integer m (1 ≤ m ≤ 1000). -----Output----- In the first line print "YES", if there is a way to put m weights on the scales by all rules. Otherwise, print in the first line "NO". If you can put m weights on the scales, then print in the next line m integers — the weights' weights in the order you put them on the scales. If there are multiple solutions, you can print any of them. -----Examples----- Input 0000000101 3 Output YES 8 10 8 Input 1000000000 2 Output NO Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. Dasha logged into the system and began to solve problems. One of them is as follows: Given two sequences a and b of length n each you need to write a sequence c of length n, the i-th element of which is calculated as follows: c_{i} = b_{i} - a_{i}. About sequences a and b we know that their elements are in the range from l to r. More formally, elements satisfy the following conditions: l ≤ a_{i} ≤ r and l ≤ b_{i} ≤ r. About sequence c we know that all its elements are distinct. [Image] Dasha wrote a solution to that problem quickly, but checking her work on the standard test was not so easy. Due to an error in the test system only the sequence a and the compressed sequence of the sequence c were known from that test. Let's give the definition to a compressed sequence. A compressed sequence of sequence c of length n is a sequence p of length n, so that p_{i} equals to the number of integers which are less than or equal to c_{i} in the sequence c. For example, for the sequence c = [250, 200, 300, 100, 50] the compressed sequence will be p = [4, 3, 5, 2, 1]. Pay attention that in c all integers are distinct. Consequently, the compressed sequence contains all integers from 1 to n inclusively. Help Dasha to find any sequence b for which the calculated compressed sequence of sequence c is correct. -----Input----- The first line contains three integers n, l, r (1 ≤ n ≤ 10^5, 1 ≤ l ≤ r ≤ 10^9) — the length of the sequence and boundaries of the segment where the elements of sequences a and b are. The next line contains n integers a_1, a_2, ..., a_{n} (l ≤ a_{i} ≤ r) — the elements of the sequence a. The next line contains n distinct integers p_1, p_2, ..., p_{n} (1 ≤ p_{i} ≤ n) — the compressed sequence of the sequence c. -----Output----- If there is no the suitable sequence b, then in the only line print "-1". Otherwise, in the only line print n integers — the elements of any suitable sequence b. -----Examples----- Input 5 1 5 1 1 1 1 1 3 1 5 4 2 Output 3 1 5 4 2 Input 4 2 9 3 4 8 9 3 2 1 4 Output 2 2 2 9 Input 6 1 5 1 1 1 1 1 1 2 3 5 4 1 6 Output -1 -----Note----- Sequence b which was found in the second sample is suitable, because calculated sequence c = [2 - 3, 2 - 4, 2 - 8, 9 - 9] = [ - 1, - 2, - 6, 0] (note that c_{i} = b_{i} - a_{i}) has compressed sequence equals to p = [3, 2, 1, 4]. Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. After you had helped George and Alex to move in the dorm, they went to help their friend Fedor play a new computer game «Call of Soldiers 3». The game has (m + 1) players and n types of soldiers in total. Players «Call of Soldiers 3» are numbered form 1 to (m + 1). Types of soldiers are numbered from 0 to n - 1. Each player has an army. Army of the i-th player can be described by non-negative integer x_{i}. Consider binary representation of x_{i}: if the j-th bit of number x_{i} equal to one, then the army of the i-th player has soldiers of the j-th type. Fedor is the (m + 1)-th player of the game. He assume that two players can become friends if their armies differ in at most k types of soldiers (in other words, binary representations of the corresponding numbers differ in at most k bits). Help Fedor and count how many players can become his friends. -----Input----- The first line contains three integers n, m, k (1 ≤ k ≤ n ≤ 20; 1 ≤ m ≤ 1000). The i-th of the next (m + 1) lines contains a single integer x_{i} (1 ≤ x_{i} ≤ 2^{n} - 1), that describes the i-th player's army. We remind you that Fedor is the (m + 1)-th player. -----Output----- Print a single integer — the number of Fedor's potential friends. -----Examples----- Input 7 3 1 8 5 111 17 Output 0 Input 3 3 3 1 2 3 4 Output 3 Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. You are given $n$ numbers $a_1, a_2, \ldots, a_n$. Is it possible to arrange them in a circle in such a way that every number is strictly less than the sum of its neighbors? For example, for the array $[1, 4, 5, 6, 7, 8]$, the arrangement on the left is valid, while arrangement on the right is not, as $5\ge 4 + 1$ and $8> 1 + 6$. [Image] -----Input----- The first line contains a single integer $n$ ($3\le n \le 10^5$) — the number of numbers. The second line contains $n$ integers $a_1, a_2, \ldots, a_n$ ($1 \leq a_i \le 10^9$) — the numbers. The given numbers are not necessarily distinct (i.e. duplicates are allowed). -----Output----- If there is no solution, output "NO" in the first line. If there is a solution, output "YES" in the first line. In the second line output $n$ numbers — elements of the array in the order they will stay in the circle. The first and the last element you output are considered neighbors in the circle. If there are multiple solutions, output any of them. You can print the circle starting with any element. -----Examples----- Input 3 2 4 3 Output YES 4 2 3 Input 5 1 2 3 4 4 Output YES 4 4 2 1 3 Input 3 13 8 5 Output NO Input 4 1 10 100 1000 Output NO -----Note----- One of the possible arrangements is shown in the first example: $4< 2 + 3$; $2 < 4 + 3$; $3< 4 + 2$. One of the possible arrangements is shown in the second example. No matter how we arrange $13, 8, 5$ in a circle in the third example, $13$ will have $8$ and $5$ as neighbors, but $13\ge 8 + 5$. There is no solution in the fourth example. Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. Ken loves ken-ken-pa (Japanese version of hopscotch). Today, he will play it on a directed graph G. G consists of N vertices numbered 1 to N, and M edges. The i-th edge points from Vertex u_i to Vertex v_i. First, Ken stands on Vertex S. He wants to reach Vertex T by repeating ken-ken-pa. In one ken-ken-pa, he does the following exactly three times: follow an edge pointing from the vertex on which he is standing. Determine if he can reach Vertex T by repeating ken-ken-pa. If the answer is yes, find the minimum number of ken-ken-pa needed to reach Vertex T. Note that visiting Vertex T in the middle of a ken-ken-pa does not count as reaching Vertex T by repeating ken-ken-pa. -----Constraints----- - 2 \leq N \leq 10^5 - 0 \leq M \leq \min(10^5, N (N-1)) - 1 \leq u_i, v_i \leq N(1 \leq i \leq M) - u_i \neq v_i (1 \leq i \leq M) - If i \neq j, (u_i, v_i) \neq (u_j, v_j). - 1 \leq S, T \leq N - S \neq T -----Input----- Input is given from Standard Input in the following format: N M u_1 v_1 : u_M v_M S T -----Output----- If Ken cannot reach Vertex T from Vertex S by repeating ken-ken-pa, print -1. If he can, print the minimum number of ken-ken-pa needed to reach vertex T. -----Sample Input----- 4 4 1 2 2 3 3 4 4 1 1 3 -----Sample Output----- 2 Ken can reach Vertex 3 from Vertex 1 in two ken-ken-pa, as follows: 1 \rightarrow 2 \rightarrow 3 \rightarrow 4 in the first ken-ken-pa, then 4 \rightarrow 1 \rightarrow 2 \rightarrow 3 in the second ken-ken-pa. This is the minimum number of ken-ken-pa needed. Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. Given are N positive integers A_1,...,A_N. Consider positive integers B_1, ..., B_N that satisfy the following condition. Condition: For any i, j such that 1 \leq i < j \leq N, A_i B_i = A_j B_j holds. Find the minimum possible value of B_1 + ... + B_N for such B_1,...,B_N. Since the answer can be enormous, print the sum modulo (10^9 +7). -----Constraints----- - 1 \leq N \leq 10^4 - 1 \leq A_i \leq 10^6 - All values in input are integers. -----Input----- Input is given from Standard Input in the following format: N A_1 ... A_N -----Output----- Print the minimum possible value of B_1 + ... + B_N for B_1,...,B_N that satisfy the condition, modulo (10^9 +7). -----Sample Input----- 3 2 3 4 -----Sample Output----- 13 Let B_1=6, B_2=4, and B_3=3, and the condition will be satisfied. Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. There is a grid with N rows and N columns of squares. Let (i,j) be the square at the i-th row from the top and the j-th column from the left. These squares have to be painted in one of the C colors from Color 1 to Color C. Initially, (i,j) is painted in Color c_{i,j}. We say the grid is a good grid when the following condition is met for all i,j,x,y satisfying 1 \leq i,j,x,y \leq N: - If (i+j) \% 3=(x+y) \% 3, the color of (i,j) and the color of (x,y) are the same. - If (i+j) \% 3 \neq (x+y) \% 3, the color of (i,j) and the color of (x,y) are different. Here, X \% Y represents X modulo Y. We will repaint zero or more squares so that the grid will be a good grid. For a square, the wrongness when the color of the square is X before repainting and Y after repainting, is D_{X,Y}. Find the minimum possible sum of the wrongness of all the squares. -----Constraints----- - 1 \leq N \leq 500 - 3 \leq C \leq 30 - 1 \leq D_{i,j} \leq 1000 (i \neq j),D_{i,j}=0 (i=j) - 1 \leq c_{i,j} \leq C - All values in input are integers. -----Input----- Input is given from Standard Input in the following format: N C D_{1,1} ... D_{1,C} : D_{C,1} ... D_{C,C} c_{1,1} ... c_{1,N} : c_{N,1} ... c_{N,N} -----Output----- If the minimum possible sum of the wrongness of all the squares is x, print x. -----Sample Input----- 2 3 0 1 1 1 0 1 1 4 0 1 2 3 3 -----Sample Output----- 3 - Repaint (1,1) to Color 2. The wrongness of (1,1) becomes D_{1,2}=1. - Repaint (1,2) to Color 3. The wrongness of (1,2) becomes D_{2,3}=1. - Repaint (2,2) to Color 1. The wrongness of (2,2) becomes D_{3,1}=1. In this case, the sum of the wrongness of all the squares is 3. Note that D_{i,j} \neq D_{j,i} is possible. Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. There are N mountains in a circle, called Mountain 1, Mountain 2, ..., Mountain N in clockwise order. N is an odd number. Between these mountains, there are N dams, called Dam 1, Dam 2, ..., Dam N. Dam i (1 \leq i \leq N) is located between Mountain i and i+1 (Mountain N+1 is Mountain 1). When Mountain i (1 \leq i \leq N) receives 2x liters of rain, Dam i-1 and Dam i each accumulates x liters of water (Dam 0 is Dam N). One day, each of the mountains received a non-negative even number of liters of rain. As a result, Dam i (1 \leq i \leq N) accumulated a total of A_i liters of water. Find the amount of rain each of the mountains received. We can prove that the solution is unique under the constraints of this problem. -----Constraints----- - All values in input are integers. - 3 \leq N \leq 10^5-1 - N is an odd number. - 0 \leq A_i \leq 10^9 - The situation represented by input can occur when each of the mountains receives a non-negative even number of liters of rain. -----Input----- Input is given from Standard Input in the following format: N A_1 A_2 ... A_N -----Output----- Print N integers representing the number of liters of rain Mountain 1, Mountain 2, ..., Mountain N received, in this order. -----Sample Input----- 3 2 2 4 -----Sample Output----- 4 0 4 If we assume Mountain 1, 2, and 3 received 4, 0, and 4 liters of rain, respectively, it is consistent with this input, as follows: - Dam 1 should have accumulated \frac{4}{2} + \frac{0}{2} = 2 liters of water. - Dam 2 should have accumulated \frac{0}{2} + \frac{4}{2} = 2 liters of water. - Dam 3 should have accumulated \frac{4}{2} + \frac{4}{2} = 4 liters of water. Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. Theater stage is a rectangular field of size n × m. The director gave you the stage's plan which actors will follow. For each cell it is stated in the plan if there would be an actor in this cell or not. You are to place a spotlight on the stage in some good position. The spotlight will project light in one of the four directions (if you look at the stage from above) — left, right, up or down. Thus, the spotlight's position is a cell it is placed to and a direction it shines. A position is good if two conditions hold: there is no actor in the cell the spotlight is placed to; there is at least one actor in the direction the spotlight projects. Count the number of good positions for placing the spotlight. Two positions of spotlight are considered to be different if the location cells or projection direction differ. -----Input----- The first line contains two positive integers n and m (1 ≤ n, m ≤ 1000) — the number of rows and the number of columns in the plan. The next n lines contain m integers, 0 or 1 each — the description of the plan. Integer 1, means there will be an actor in the corresponding cell, while 0 means the cell will remain empty. It is guaranteed that there is at least one actor in the plan. -----Output----- Print one integer — the number of good positions for placing the spotlight. -----Examples----- Input 2 4 0 1 0 0 1 0 1 0 Output 9 Input 4 4 0 0 0 0 1 0 0 1 0 1 1 0 0 1 0 0 Output 20 -----Note----- In the first example the following positions are good: the (1, 1) cell and right direction; the (1, 1) cell and down direction; the (1, 3) cell and left direction; the (1, 3) cell and down direction; the (1, 4) cell and left direction; the (2, 2) cell and left direction; the (2, 2) cell and up direction; the (2, 2) and right direction; the (2, 4) cell and left direction. Therefore, there are 9 good positions in this example. Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. Let's define a forest as a non-directed acyclic graph (also without loops and parallel edges). One day Misha played with the forest consisting of n vertices. For each vertex v from 0 to n - 1 he wrote down two integers, degree_{v} and s_{v}, were the first integer is the number of vertices adjacent to vertex v, and the second integer is the XOR sum of the numbers of vertices adjacent to v (if there were no adjacent vertices, he wrote down 0). Next day Misha couldn't remember what graph he initially had. Misha has values degree_{v} and s_{v} left, though. Help him find the number of edges and the edges of the initial graph. It is guaranteed that there exists a forest that corresponds to the numbers written by Misha. -----Input----- The first line contains integer n (1 ≤ n ≤ 2^16), the number of vertices in the graph. The i-th of the next lines contains numbers degree_{i} and s_{i} (0 ≤ degree_{i} ≤ n - 1, 0 ≤ s_{i} < 2^16), separated by a space. -----Output----- In the first line print number m, the number of edges of the graph. Next print m lines, each containing two distinct numbers, a and b (0 ≤ a ≤ n - 1, 0 ≤ b ≤ n - 1), corresponding to edge (a, b). Edges can be printed in any order; vertices of the edge can also be printed in any order. -----Examples----- Input 3 2 3 1 0 1 0 Output 2 1 0 2 0 Input 2 1 1 1 0 Output 1 0 1 -----Note----- The XOR sum of numbers is the result of bitwise adding numbers modulo 2. This operation exists in many modern programming languages. For example, in languages C++, Java and Python it is represented as "^", and in Pascal — as "xor". Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. Inna loves digit 9 very much. That's why she asked Dima to write a small number consisting of nines. But Dima must have misunderstood her and he wrote a very large number a, consisting of digits from 1 to 9. Inna wants to slightly alter the number Dima wrote so that in the end the number contained as many digits nine as possible. In one move, Inna can choose two adjacent digits in a number which sum equals 9 and replace them by a single digit 9. For instance, Inna can alter number 14545181 like this: 14545181 → 1945181 → 194519 → 19919. Also, she can use this method to transform number 14545181 into number 19991. Inna will not transform it into 149591 as she can get numbers 19919 and 19991 which contain more digits nine. Dima is a programmer so he wants to find out how many distinct numbers containing as many digits nine as possible Inna can get from the written number. Help him with this challenging task. -----Input----- The first line of the input contains integer a (1 ≤ a ≤ 10^100000). Number a doesn't have any zeroes. -----Output----- In a single line print a single integer — the answer to the problem. It is guaranteed that the answer to the problem doesn't exceed 2^63 - 1. Please, do not use the %lld specifier to read or write 64-bit integers in С++. It is preferred to use the cin, cout streams or the %I64d specifier. -----Examples----- Input 369727 Output 2 Input 123456789987654321 Output 1 Input 1 Output 1 -----Note----- Notes to the samples In the first sample Inna can get the following numbers: 369727 → 99727 → 9997, 369727 → 99727 → 9979. In the second sample, Inna can act like this: 123456789987654321 → 12396789987654321 → 1239678998769321. Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. The police department of your city has just started its journey. Initially, they don’t have any manpower. So, they started hiring new recruits in groups. Meanwhile, crimes keeps occurring within the city. One member of the police force can investigate only one crime during his/her lifetime. If there is no police officer free (isn't busy with crime) during the occurrence of a crime, it will go untreated. Given the chronological order of crime occurrences and recruit hirings, find the number of crimes which will go untreated. -----Input----- The first line of input will contain an integer n (1 ≤ n ≤ 10^5), the number of events. The next line will contain n space-separated integers. If the integer is -1 then it means a crime has occurred. Otherwise, the integer will be positive, the number of officers recruited together at that time. No more than 10 officers will be recruited at a time. -----Output----- Print a single integer, the number of crimes which will go untreated. -----Examples----- Input 3 -1 -1 1 Output 2 Input 8 1 -1 1 -1 -1 1 1 1 Output 1 Input 11 -1 -1 2 -1 -1 -1 -1 -1 -1 -1 -1 Output 8 -----Note----- Lets consider the second example: Firstly one person is hired. Then crime appears, the last hired person will investigate this crime. One more person is hired. One more crime appears, the last hired person will investigate this crime. Crime appears. There is no free policeman at the time, so this crime will go untreated. One more person is hired. One more person is hired. One more person is hired. The answer is one, as one crime (on step 5) will go untreated. Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. While walking down the street Vanya saw a label "Hide&Seek". Because he is a programmer, he used & as a bitwise AND for these two words represented as a integers in base 64 and got new word. Now Vanya thinks of some string s and wants to know the number of pairs of words of length \|s\| (length of s), such that their bitwise AND is equal to s. As this number can be large, output it modulo 10^9 + 7. To represent the string as a number in numeral system with base 64 Vanya uses the following rules: digits from '0' to '9' correspond to integers from 0 to 9; letters from 'A' to 'Z' correspond to integers from 10 to 35; letters from 'a' to 'z' correspond to integers from 36 to 61; letter '-' correspond to integer 62; letter '_' correspond to integer 63. -----Input----- The only line of the input contains a single word s (1 ≤ \|s\| ≤ 100 000), consisting of digits, lowercase and uppercase English letters, characters '-' and '_'. -----Output----- Print a single integer — the number of possible pairs of words, such that their bitwise AND is equal to string s modulo 10^9 + 7. -----Examples----- Input z Output 3 Input V_V Output 9 Input Codeforces Output 130653412 -----Note----- For a detailed definition of bitwise AND we recommend to take a look in the corresponding article in Wikipedia. In the first sample, there are 3 possible solutions: z&_ = 61&63 = 61 = z _&z = 63&61 = 61 = z z&z = 61&61 = 61 = z Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. Pavel has several sticks with lengths equal to powers of two. He has $a_0$ sticks of length $2^0 = 1$, $a_1$ sticks of length $2^1 = 2$, ..., $a_{n-1}$ sticks of length $2^{n-1}$. Pavel wants to make the maximum possible number of triangles using these sticks. The triangles should have strictly positive area, each stick can be used in at most one triangle. It is forbidden to break sticks, and each triangle should consist of exactly three sticks. Find the maximum possible number of triangles. -----Input----- The first line contains a single integer $n$ ($1 \leq n \leq 300\,000$) — the number of different lengths of sticks. The second line contains $n$ integers $a_0$, $a_1$, ..., $a_{n-1}$ ($1 \leq a_i \leq 10^9$), where $a_i$ is the number of sticks with the length equal to $2^i$. -----Output----- Print a single integer — the maximum possible number of non-degenerate triangles that Pavel can make. -----Examples----- Input 5 1 2 2 2 2 Output 3 Input 3 1 1 1 Output 0 Input 3 3 3 3 Output 3 -----Note----- In the first example, Pavel can, for example, make this set of triangles (the lengths of the sides of the triangles are listed): $(2^0, 2^4, 2^4)$, $(2^1, 2^3, 2^3)$, $(2^1, 2^2, 2^2)$. In the second example, Pavel cannot make a single triangle. In the third example, Pavel can, for example, create this set of triangles (the lengths of the sides of the triangles are listed): $(2^0, 2^0, 2^0)$, $(2^1, 2^1, 2^1)$, $(2^2, 2^2, 2^2)$. Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. Smart Beaver decided to be not only smart, but also a healthy beaver! And so he began to attend physical education classes at school X. In this school, physical education has a very creative teacher. One of his favorite warm-up exercises is throwing balls. Students line up. Each one gets a single ball in the beginning. The balls are numbered from 1 to n (by the demand of the inventory commission). [Image] Figure 1. The initial position for n = 5. After receiving the balls the students perform the warm-up exercise. The exercise takes place in a few throws. For each throw the teacher chooses any two arbitrary different students who will participate in it. The selected students throw their balls to each other. Thus, after each throw the students remain in their positions, and the two balls are swapped. [Image] Figure 2. The example of a throw. In this case there was a throw between the students, who were holding the 2-nd and the 4-th balls. Since the warm-up has many exercises, each of them can only continue for little time. Therefore, for each student we know the maximum number of throws he can participate in. For this lessons maximum number of throws will be 1 or 2. Note that after all phases of the considered exercise any ball can end up with any student. Smart Beaver decided to formalize it and introduced the concept of the "ball order". The ball order is a sequence of n numbers that correspond to the order of balls in the line. The first number will match the number of the ball of the first from the left student in the line, the second number will match the ball of the second student, and so on. For example, in figure 2 the order of the balls was (1, 2, 3, 4, 5), and after the throw it was (1, 4, 3, 2, 5). Smart beaver knows the number of students and for each student he knows the maximum number of throws in which he can participate. And now he is wondering: what is the number of distinct ways of ball orders by the end of the exercise. -----Input----- The first line contains a single number n — the number of students in the line and the number of balls. The next line contains exactly n space-separated integers. Each number corresponds to a student in the line (the i-th number corresponds to the i-th from the left student in the line) and shows the number of throws he can participate in. The input limits for scoring 30 points are (subproblem D1): 1 ≤ n ≤ 10. The input limits for scoring 70 points are (subproblems D1+D2): 1 ≤ n ≤ 500. The input limits for scoring 100 points are (subproblems D1+D2+D3): 1 ≤ n ≤ 1000000. -----Output----- The output should contain a single integer — the number of variants of ball orders after the warm up exercise is complete. As the number can be rather large, print it modulo 1000000007 (10^9 + 7). -----Examples----- Input 5 1 2 2 1 2 Output 120 Input 8 1 2 2 1 2 1 1 2 Output 16800 Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. In this task you need to process a set of stock exchange orders and use them to create order book. An order is an instruction of some participant to buy or sell stocks on stock exchange. The order number i has price p_{i}, direction d_{i} — buy or sell, and integer q_{i}. This means that the participant is ready to buy or sell q_{i} stocks at price p_{i} for one stock. A value q_{i} is also known as a volume of an order. All orders with the same price p and direction d are merged into one aggregated order with price p and direction d. The volume of such order is a sum of volumes of the initial orders. An order book is a list of aggregated orders, the first part of which contains sell orders sorted by price in descending order, the second contains buy orders also sorted by price in descending order. An order book of depth s contains s best aggregated orders for each direction. A buy order is better if it has higher price and a sell order is better if it has lower price. If there are less than s aggregated orders for some direction then all of them will be in the final order book. You are given n stock exhange orders. Your task is to print order book of depth s for these orders. -----Input----- The input starts with two positive integers n and s (1 ≤ n ≤ 1000, 1 ≤ s ≤ 50), the number of orders and the book depth. Next n lines contains a letter d_{i} (either 'B' or 'S'), an integer p_{i} (0 ≤ p_{i} ≤ 10^5) and an integer q_{i} (1 ≤ q_{i} ≤ 10^4) — direction, price and volume respectively. The letter 'B' means buy, 'S' means sell. The price of any sell order is higher than the price of any buy order. -----Output----- Print no more than 2s lines with aggregated orders from order book of depth s. The output format for orders should be the same as in input. -----Examples----- Input 6 2 B 10 3 S 50 2 S 40 1 S 50 6 B 20 4 B 25 10 Output S 50 8 S 40 1 B 25 10 B 20 4 -----Note----- Denote (x, y) an order with price x and volume y. There are 3 aggregated buy orders (10, 3), (20, 4), (25, 10) and two sell orders (50, 8), (40, 1) in the sample. You need to print no more than two best orders for each direction, so you shouldn't print the order (10 3) having the worst price among buy orders. Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. Bob is a duck. He wants to get to Alice's nest, so that those two can duck! [Image] Duck is the ultimate animal! (Image courtesy of See Bang) The journey can be represented as a straight line, consisting of $n$ segments. Bob is located to the left of the first segment, while Alice's nest is on the right of the last segment. Each segment has a length in meters, and also terrain type: grass, water or lava. Bob has three movement types: swimming, walking and flying. He can switch between them or change his direction at any point in time (even when he is located at a non-integer coordinate), and doing so doesn't require any extra time. Bob can swim only on the water, walk only on the grass and fly over any terrain. Flying one meter takes $1$ second, swimming one meter takes $3$ seconds, and finally walking one meter takes $5$ seconds. Bob has a finite amount of energy, called stamina. Swimming and walking is relaxing for him, so he gains $1$ stamina for every meter he walks or swims. On the other hand, flying is quite tiring, and he spends $1$ stamina for every meter flown. Staying in place does not influence his stamina at all. Of course, his stamina can never become negative. Initially, his stamina is zero. What is the shortest possible time in which he can reach Alice's nest? -----Input----- The first line contains a single integer $n$ ($1 \leq n \leq 10^5$) — the number of segments of terrain. The second line contains $n$ integers $l_1, l_2, \dots, l_n$ ($1 \leq l_i \leq 10^{12}$). The $l_i$ represents the length of the $i$-th terrain segment in meters. The third line contains a string $s$ consisting of $n$ characters "G", "W", "L", representing Grass, Water and Lava, respectively. It is guaranteed that the first segment is not Lava. -----Output----- Output a single integer $t$ — the minimum time Bob needs to reach Alice. -----Examples----- Input 1 10 G Output 30 Input 2 10 10 WL Output 40 Input 2 1 2 WL Output 8 Input 3 10 10 10 GLW Output 80 -----Note----- In the first sample, Bob first walks $5$ meters in $25$ seconds. Then he flies the remaining $5$ meters in $5$ seconds. In the second sample, Bob first swims $10$ meters in $30$ seconds. Then he flies over the patch of lava for $10$ seconds. In the third sample, the water pond is much smaller. Bob first swims over the water pond, taking him $3$ seconds. However, he cannot fly over the lava just yet, as he only has one stamina while he needs two. So he swims back for half a meter, and then half a meter forward, taking him $3$ seconds in total. Now he has $2$ stamina, so he can spend $2$ seconds flying over the lava. In the fourth sample, he walks for $50$ seconds, flies for $10$ seconds, swims for $15$ seconds, and finally flies for $5$ seconds. Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. Berland is going through tough times — the dirt price has dropped and that is a blow to the country's economy. Everybody knows that Berland is the top world dirt exporter! The President of Berland was forced to leave only k of the currently existing n subway stations. The subway stations are located on a straight line one after another, the trains consecutively visit the stations as they move. You can assume that the stations are on the Ox axis, the i-th station is at point with coordinate x_{i}. In such case the distance between stations i and j is calculated by a simple formula \|x_{i} - x_{j}\|. Currently, the Ministry of Transport is choosing which stations to close and which ones to leave. Obviously, the residents of the capital won't be too enthusiastic about the innovation, so it was decided to show the best side to the people. The Ministry of Transport wants to choose such k stations that minimize the average commute time in the subway! Assuming that the train speed is constant (it is a fixed value), the average commute time in the subway is calculated as the sum of pairwise distances between stations, divided by the number of pairs (that is $\frac{n \cdot(n - 1)}{2}$) and divided by the speed of the train. Help the Minister of Transport to solve this difficult problem. Write a program that, given the location of the stations selects such k stations that the average commute time in the subway is minimized. -----Input----- The first line of the input contains integer n (3 ≤ n ≤ 3·10^5) — the number of the stations before the innovation. The second line contains the coordinates of the stations x_1, x_2, ..., x_{n} ( - 10^8 ≤ x_{i} ≤ 10^8). The third line contains integer k (2 ≤ k ≤ n - 1) — the number of stations after the innovation. The station coordinates are distinct and not necessarily sorted. -----Output----- Print a sequence of k distinct integers t_1, t_2, ..., t_{k} (1 ≤ t_{j} ≤ n) — the numbers of the stations that should be left after the innovation in arbitrary order. Assume that the stations are numbered 1 through n in the order they are given in the input. The number of stations you print must have the minimum possible average commute time among all possible ways to choose k stations. If there are multiple such ways, you are allowed to print any of them. -----Examples----- Input 3 1 100 101 2 Output 2 3 -----Note----- In the sample testcase the optimal answer is to destroy the first station (with x = 1). The average commute time will be equal to 1 in this way. Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. Dima took up the biology of bacteria, as a result of his experiments, he invented k types of bacteria. Overall, there are n bacteria at his laboratory right now, and the number of bacteria of type i equals c_{i}. For convenience, we will assume that all the bacteria are numbered from 1 to n. The bacteria of type c_{i} are numbered from $(\sum_{k = 1}^{i - 1} c_{k}) + 1$ to $\sum_{k = 1}^{i} c_{k}$. With the help of special equipment Dima can move energy from some bacteria into some other one. Of course, the use of such equipment is not free. Dima knows m ways to move energy from some bacteria to another one. The way with number i can be described with integers u_{i}, v_{i} and x_{i} mean that this way allows moving energy from bacteria with number u_{i} to bacteria with number v_{i} or vice versa for x_{i} dollars. Dima's Chef (Inna) calls the type-distribution correct if there is a way (may be non-direct) to move energy from any bacteria of the particular type to any other bacteria of the same type (between any two bacteria of the same type) for zero cost. As for correct type-distribution the cost of moving the energy depends only on the types of bacteria help Inna to determine is the type-distribution correct? If it is, print the matrix d with size k × k. Cell d[i][j] of this matrix must be equal to the minimal possible cost of energy-moving from bacteria with type i to bacteria with type j. -----Input----- The first line contains three integers n, m, k (1 ≤ n ≤ 10^5; 0 ≤ m ≤ 10^5; 1 ≤ k ≤ 500). The next line contains k integers c_1, c_2, ..., c_{k} (1 ≤ c_{i} ≤ n). Each of the next m lines contains three integers u_{i}, v_{i}, x_{i} (1 ≤ u_{i}, v_{i} ≤ 10^5; 0 ≤ x_{i} ≤ 10^4). It is guaranteed that $\sum_{i = 1}^{k} c_{i} = n$. -----Output----- If Dima's type-distribution is correct, print string «Yes», and then k lines: in the i-th line print integers d[i][1], d[i][2], ..., d[i][k] (d[i][i] = 0). If there is no way to move energy from bacteria i to bacteria j appropriate d[i][j] must equal to -1. If the type-distribution isn't correct print «No». -----Examples----- Input 4 4 2 1 3 2 3 0 3 4 0 2 4 1 2 1 2 Output Yes 0 2 2 0 Input 3 1 2 2 1 1 2 0 Output Yes 0 -1 -1 0 Input 3 2 2 2 1 1 2 0 2 3 1 Output Yes 0 1 1 0 Input 3 0 2 1 2 Output No Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. Alex enjoys performing magic tricks. He has a trick that requires a deck of n cards. He has m identical decks of n different cards each, which have been mixed together. When Alex wishes to perform the trick, he grabs n cards at random and performs the trick with those. The resulting deck looks like a normal deck, but may have duplicates of some cards. The trick itself is performed as follows: first Alex allows you to choose a random card from the deck. You memorize the card and put it back in the deck. Then Alex shuffles the deck, and pulls out a card. If the card matches the one you memorized, the trick is successful. You don't think Alex is a very good magician, and that he just pulls a card randomly from the deck. Determine the probability of the trick being successful if this is the case. -----Input----- First line of the input consists of two integers n and m (1 ≤ n, m ≤ 1000), separated by space — number of cards in each deck, and number of decks. -----Output----- On the only line of the output print one floating point number – probability of Alex successfully performing the trick. Relative or absolute error of your answer should not be higher than 10^{ - 6}. -----Examples----- Input 2 2 Output 0.6666666666666666 Input 4 4 Output 0.4000000000000000 Input 1 2 Output 1.0000000000000000 -----Note----- In the first sample, with probability $\frac{1}{3}$ Alex will perform the trick with two cards with the same value from two different decks. In this case the trick is guaranteed to succeed. With the remaining $\frac{2}{3}$ probability he took two different cards, and the probability of pulling off the trick is $\frac{1}{2}$. The resulting probability is $\frac{1}{3} \times 1 + \frac{2}{3} \times \frac{1}{2} = \frac{2}{3}$ Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. Vasya owns a cornfield which can be defined with two integers $n$ and $d$. The cornfield can be represented as rectangle with vertices having Cartesian coordinates $(0, d), (d, 0), (n, n - d)$ and $(n - d, n)$. [Image] An example of a cornfield with $n = 7$ and $d = 2$. Vasya also knows that there are $m$ grasshoppers near the field (maybe even inside it). The $i$-th grasshopper is at the point $(x_i, y_i)$. Vasya does not like when grasshoppers eat his corn, so for each grasshopper he wants to know whether its position is inside the cornfield (including the border) or outside. Help Vasya! For each grasshopper determine if it is inside the field (including the border). -----Input----- The first line contains two integers $n$ and $d$ ($1 \le d < n \le 100$). The second line contains a single integer $m$ ($1 \le m \le 100$) — the number of grasshoppers. The $i$-th of the next $m$ lines contains two integers $x_i$ and $y_i$ ($0 \le x_i, y_i \le n$) — position of the $i$-th grasshopper. -----Output----- Print $m$ lines. The $i$-th line should contain "YES" if the position of the $i$-th grasshopper lies inside or on the border of the cornfield. Otherwise the $i$-th line should contain "NO". You can print each letter in any case (upper or lower). -----Examples----- Input 7 2 4 2 4 4 1 6 3 4 5 Output YES NO NO YES Input 8 7 4 4 4 2 8 8 1 6 1 Output YES NO YES YES -----Note----- The cornfield from the first example is pictured above. Grasshoppers with indices $1$ (coordinates $(2, 4)$) and $4$ (coordinates $(4, 5)$) are inside the cornfield. The cornfield from the second example is pictured below. Grasshoppers with indices $1$ (coordinates $(4, 4)$), $3$ (coordinates $(8, 1)$) and $4$ (coordinates $(6, 1)$) are inside the cornfield. [Image] Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. You are given a non-decreasing array of non-negative integers $a_1, a_2, \ldots, a_n$. Also you are given a positive integer $k$. You want to find $m$ non-decreasing arrays of non-negative integers $b_1, b_2, \ldots, b_m$, such that: The size of $b_i$ is equal to $n$ for all $1 \leq i \leq m$. For all $1 \leq j \leq n$, $a_j = b_{1, j} + b_{2, j} + \ldots + b_{m, j}$. In the other word, array $a$ is the sum of arrays $b_i$. The number of different elements in the array $b_i$ is at most $k$ for all $1 \leq i \leq m$. Find the minimum possible value of $m$, or report that there is no possible $m$. -----Input----- The first line contains one integer $t$ ($1 \leq t \leq 100$): the number of test cases. The first line of each test case contains two integers $n$, $k$ ($1 \leq n \leq 100$, $1 \leq k \leq n$). The second line contains $n$ integers $a_1, a_2, \ldots, a_n$ ($0 \leq a_1 \leq a_2 \leq \ldots \leq a_n \leq 100$, $a_n > 0$). -----Output----- For each test case print a single integer: the minimum possible value of $m$. If there is no such $m$, print $-1$. -----Example----- Input 6 4 1 0 0 0 1 3 1 3 3 3 11 3 0 1 2 2 3 3 3 4 4 4 4 5 3 1 2 3 4 5 9 4 2 2 3 5 7 11 13 13 17 10 7 0 1 1 2 3 3 4 5 5 6 Output -1 1 2 2 2 1 -----Note----- In the first test case, there is no possible $m$, because all elements of all arrays should be equal to $0$. But in this case, it is impossible to get $a_4 = 1$ as the sum of zeros. In the second test case, we can take $b_1 = [3, 3, 3]$. $1$ is the smallest possible value of $m$. In the third test case, we can take $b_1 = [0, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2]$ and $b_2 = [0, 0, 1, 1, 1, 1, 1, 2, 2, 2, 2]$. It's easy to see, that $a_i = b_{1, i} + b_{2, i}$ for all $i$ and the number of different elements in $b_1$ and in $b_2$ is equal to $3$ (so it is at most $3$). It can be proven that $2$ is the smallest possible value of $m$. Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. Suppose there is a $h \times w$ grid consisting of empty or full cells. Let's make some definitions: $r_{i}$ is the number of consecutive full cells connected to the left side in the $i$-th row ($1 \le i \le h$). In particular, $r_i=0$ if the leftmost cell of the $i$-th row is empty. $c_{j}$ is the number of consecutive full cells connected to the top end in the $j$-th column ($1 \le j \le w$). In particular, $c_j=0$ if the topmost cell of the $j$-th column is empty. In other words, the $i$-th row starts exactly with $r_i$ full cells. Similarly, the $j$-th column starts exactly with $c_j$ full cells. [Image] These are the $r$ and $c$ values of some $3 \times 4$ grid. Black cells are full and white cells are empty. You have values of $r$ and $c$. Initially, all cells are empty. Find the number of ways to fill grid cells to satisfy values of $r$ and $c$. Since the answer can be very large, find the answer modulo $1000000007\,(10^{9} + 7)$. In other words, find the remainder after division of the answer by $1000000007\,(10^{9} + 7)$. -----Input----- The first line contains two integers $h$ and $w$ ($1 \le h, w \le 10^{3}$) — the height and width of the grid. The second line contains $h$ integers $r_{1}, r_{2}, \ldots, r_{h}$ ($0 \le r_{i} \le w$) — the values of $r$. The third line contains $w$ integers $c_{1}, c_{2}, \ldots, c_{w}$ ($0 \le c_{j} \le h$) — the values of $c$. -----Output----- Print the answer modulo $1000000007\,(10^{9} + 7)$. -----Examples----- Input 3 4 0 3 1 0 2 3 0 Output 2 Input 1 1 0 1 Output 0 Input 19 16 16 16 16 16 15 15 0 5 0 4 9 9 1 4 4 0 8 16 12 6 12 19 15 8 6 19 19 14 6 9 16 10 11 15 4 Output 797922655 -----Note----- In the first example, this is the other possible case. [Image] In the second example, it's impossible to make a grid to satisfy such $r$, $c$ values. In the third example, make sure to print answer modulo $(10^9 + 7)$. Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. Tsumugi brought $n$ delicious sweets to the Light Music Club. They are numbered from $1$ to $n$, where the $i$-th sweet has a sugar concentration described by an integer $a_i$. Yui loves sweets, but she can eat at most $m$ sweets each day for health reasons. Days are $1$-indexed (numbered $1, 2, 3, \ldots$). Eating the sweet $i$ at the $d$-th day will cause a sugar penalty of $(d \cdot a_i)$, as sweets become more sugary with time. A sweet can be eaten at most once. The total sugar penalty will be the sum of the individual penalties of each sweet eaten. Suppose that Yui chooses exactly $k$ sweets, and eats them in any order she wants. What is the minimum total sugar penalty she can get? Since Yui is an undecided girl, she wants you to answer this question for every value of $k$ between $1$ and $n$. -----Input----- The first line contains two integers $n$ and $m$ ($1 \le m \le n \le 200\ 000$). The second line contains $n$ integers $a_1, a_2, \ldots, a_n$ ($1 \le a_i \le 200\ 000$). -----Output----- You have to output $n$ integers $x_1, x_2, \ldots, x_n$ on a single line, separed by spaces, where $x_k$ is the minimum total sugar penalty Yui can get if she eats exactly $k$ sweets. -----Examples----- Input 9 2 6 19 3 4 4 2 6 7 8 Output 2 5 11 18 30 43 62 83 121 Input 1 1 7 Output 7 -----Note----- Let's analyze the answer for $k = 5$ in the first example. Here is one of the possible ways to eat $5$ sweets that minimize total sugar penalty: Day $1$: sweets $1$ and $4$ Day $2$: sweets $5$ and $3$ Day $3$ : sweet $6$ Total penalty is $1 \cdot a_1 + 1 \cdot a_4 + 2 \cdot a_5 + 2 \cdot a_3 + 3 \cdot a_6 = 6 + 4 + 8 + 6 + 6 = 30$. We can prove that it's the minimum total sugar penalty Yui can achieve if she eats $5$ sweets, hence $x_5 = 30$. Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. In this problem, a $n \times m$ rectangular matrix $a$ is called increasing if, for each row of $i$, when go from left to right, the values strictly increase (that is, $a_{i,1}<a_{i,2}<\dots<a_{i,m}$) and for each column $j$, when go from top to bottom, the values strictly increase (that is, $a_{1,j}<a_{2,j}<\dots<a_{n,j}$). In a given matrix of non-negative integers, it is necessary to replace each value of $0$ with some positive integer so that the resulting matrix is increasing and the sum of its elements is maximum, or find that it is impossible. It is guaranteed that in a given value matrix all values of $0$ are contained only in internal cells (that is, not in the first or last row and not in the first or last column). -----Input----- The first line contains integers $n$ and $m$ ($3 \le n, m \le 500$) — the number of rows and columns in the given matrix $a$. The following lines contain $m$ each of non-negative integers — the values in the corresponding row of the given matrix: $a_{i,1}, a_{i,2}, \dots, a_{i,m}$ ($0 \le a_{i,j} \le 8000$). It is guaranteed that for all $a_{i,j}=0$, $1 < i < n$ and $1 < j < m$ are true. -----Output----- If it is possible to replace all zeros with positive numbers so that the matrix is increasing, print the maximum possible sum of matrix elements. Otherwise, print -1. -----Examples----- Input 4 5 1 3 5 6 7 3 0 7 0 9 5 0 0 0 10 8 9 10 11 12 Output 144 Input 3 3 1 2 3 2 0 4 4 5 6 Output 30 Input 3 3 1 2 3 3 0 4 4 5 6 Output -1 Input 3 3 1 2 3 2 3 4 3 4 2 Output -1 -----Note----- In the first example, the resulting matrix is as follows: 1 3 5 6 7 3 6 7 8 9 5 7 8 9 10 8 9 10 11 12 In the second example, the value $3$ must be put in the middle cell. In the third example, the desired resultant matrix does not exist. Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. Nauuo is a girl who loves playing chess. One day she invented a game by herself which needs $n$ chess pieces to play on a $m\times m$ chessboard. The rows and columns are numbered from $1$ to $m$. We denote a cell on the intersection of the $r$-th row and $c$-th column as $(r,c)$. The game's goal is to place $n$ chess pieces numbered from $1$ to $n$ on the chessboard, the $i$-th piece lies on $(r_i,\,c_i)$, while the following rule is satisfied: for all pairs of pieces $i$ and $j$, $\|r_i-r_j\|+\|c_i-c_j\|\ge\|i-j\|$. Here $\|x\|$ means the absolute value of $x$. However, Nauuo discovered that sometimes she couldn't find a solution because the chessboard was too small. She wants to find the smallest chessboard on which she can put $n$ pieces according to the rules. She also wonders how to place the pieces on such a chessboard. Can you help her? -----Input----- The only line contains a single integer $n$ ($1\le n\le 1000$) — the number of chess pieces for the game. -----Output----- The first line contains a single integer — the minimum value of $m$, where $m$ is the length of sides of the suitable chessboard. The $i$-th of the next $n$ lines contains two integers $r_i$ and $c_i$ ($1\le r_i,c_i\le m$) — the coordinates of the $i$-th chess piece. If there are multiple answers, print any. -----Examples----- Input 2 Output 2 1 1 1 2 Input 4 Output 3 1 1 1 3 3 1 3 3 -----Note----- In the first example, you can't place the two pieces on a $1\times1$ chessboard without breaking the rule. But you can place two pieces on a $2\times2$ chessboard like this: [Image] In the second example, you can't place four pieces on a $2\times2$ chessboard without breaking the rule. For example, if you place the pieces like this: [Image] then $\|r_1-r_3\|+\|c_1-c_3\|=\|1-2\|+\|1-1\|=1$, $\|1-3\|=2$, $1<2$; and $\|r_1-r_4\|+\|c_1-c_4\|=\|1-2\|+\|1-2\|=2$, $\|1-4\|=3$, $2<3$. It doesn't satisfy the rule. However, on a $3\times3$ chessboard, you can place four pieces like this: [Image] Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. Igor has been into chess for a long time and now he is sick of the game by the ordinary rules. He is going to think of new rules of the game and become world famous. Igor's chessboard is a square of size n × n cells. Igor decided that simple rules guarantee success, that's why his game will have only one type of pieces. Besides, all pieces in his game are of the same color. The possible moves of a piece are described by a set of shift vectors. The next passage contains a formal description of available moves. Let the rows of the board be numbered from top to bottom and the columns be numbered from left to right from 1 to n. Let's assign to each square a pair of integers (x, y) — the number of the corresponding column and row. Each of the possible moves of the piece is defined by a pair of integers (dx, dy); using this move, the piece moves from the field (x, y) to the field (x + dx, y + dy). You can perform the move if the cell (x + dx, y + dy) is within the boundaries of the board and doesn't contain another piece. Pieces that stand on the cells other than (x, y) and (x + dx, y + dy) are not important when considering the possibility of making the given move (for example, like when a knight moves in usual chess). Igor offers you to find out what moves his chess piece can make. He placed several pieces on the board and for each unoccupied square he told you whether it is attacked by any present piece (i.e. whether some of the pieces on the field can move to that cell). Restore a possible set of shift vectors of the piece, or else determine that Igor has made a mistake and such situation is impossible for any set of shift vectors. -----Input----- The first line contains a single integer n (1 ≤ n ≤ 50). The next n lines contain n characters each describing the position offered by Igor. The j-th character of the i-th string can have the following values: o — in this case the field (i, j) is occupied by a piece and the field may or may not be attacked by some other piece; x — in this case field (i, j) is attacked by some piece; . — in this case field (i, j) isn't attacked by any piece. It is guaranteed that there is at least one piece on the board. -----Output----- If there is a valid set of moves, in the first line print a single word 'YES' (without the quotes). Next, print the description of the set of moves of a piece in the form of a (2n - 1) × (2n - 1) board, the center of the board has a piece and symbols 'x' mark cells that are attacked by it, in a format similar to the input. See examples of the output for a full understanding of the format. If there are several possible answers, print any of them. If a valid set of moves does not exist, print a single word 'NO'. -----Examples----- Input 5 oxxxx x...x x...x x...x xxxxo Output YES ....x.... ....x.... ....x.... ....x.... xxxxoxxxx ....x.... ....x.... ....x.... ....x.... Input 6 .x.x.. x.x.x. .xo..x x..ox. .x.x.x ..x.x. Output YES ........... ........... ........... ....x.x.... ...x...x... .....o..... ...x...x... ....x.x.... ........... ........... ........... Input 3 o.x oxx o.x Output NO -----Note----- In the first sample test the piece is a usual chess rook, and in the second sample test the piece is a usual chess knight. Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. A long time ago, in a galaxy far far away two giant IT-corporations Pineapple and Gogol continue their fierce competition. Crucial moment is just around the corner: Gogol is ready to release it's new tablet Lastus 3000. This new device is equipped with specially designed artificial intelligence (AI). Employees of Pineapple did their best to postpone the release of Lastus 3000 as long as possible. Finally, they found out, that the name of the new artificial intelligence is similar to the name of the phone, that Pineapple released 200 years ago. As all rights on its name belong to Pineapple, they stand on changing the name of Gogol's artificial intelligence. Pineapple insists, that the name of their phone occurs in the name of AI as a substring. Because the name of technology was already printed on all devices, the Gogol's director decided to replace some characters in AI name with "#". As this operation is pretty expensive, you should find the minimum number of characters to replace with "#", such that the name of AI doesn't contain the name of the phone as a substring. Substring is a continuous subsequence of a string. -----Input----- The first line of the input contains the name of AI designed by Gogol, its length doesn't exceed 100 000 characters. Second line contains the name of the phone released by Pineapple 200 years ago, its length doesn't exceed 30. Both string are non-empty and consist of only small English letters. -----Output----- Print the minimum number of characters that must be replaced with "#" in order to obtain that the name of the phone doesn't occur in the name of AI as a substring. -----Examples----- Input intellect tell Output 1 Input google apple Output 0 Input sirisiri sir Output 2 -----Note----- In the first sample AI's name may be replaced with "int#llect". In the second sample Gogol can just keep things as they are. In the third sample one of the new possible names of AI may be "s#ris#ri". Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. The circle line of the Berland subway has n stations. We know the distances between all pairs of neighboring stations: d_1 is the distance between the 1-st and the 2-nd station; d_2 is the distance between the 2-nd and the 3-rd station; ... d_{n} - 1 is the distance between the n - 1-th and the n-th station; d_{n} is the distance between the n-th and the 1-st station. The trains go along the circle line in both directions. Find the shortest distance between stations with numbers s and t. -----Input----- The first line contains integer n (3 ≤ n ≤ 100) — the number of stations on the circle line. The second line contains n integers d_1, d_2, ..., d_{n} (1 ≤ d_{i} ≤ 100) — the distances between pairs of neighboring stations. The third line contains two integers s and t (1 ≤ s, t ≤ n) — the numbers of stations, between which you need to find the shortest distance. These numbers can be the same. The numbers in the lines are separated by single spaces. -----Output----- Print a single number — the length of the shortest path between stations number s and t. -----Examples----- Input 4 2 3 4 9 1 3 Output 5 Input 4 5 8 2 100 4 1 Output 15 Input 3 1 1 1 3 1 Output 1 Input 3 31 41 59 1 1 Output 0 -----Note----- In the first sample the length of path 1 → 2 → 3 equals 5, the length of path 1 → 4 → 3 equals 13. In the second sample the length of path 4 → 1 is 100, the length of path 4 → 3 → 2 → 1 is 15. In the third sample the length of path 3 → 1 is 1, the length of path 3 → 2 → 1 is 2. In the fourth sample the numbers of stations are the same, so the shortest distance equals 0. Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. Leo Jr. draws pictures in his notebook with checkered sheets (that is, each sheet has a regular square grid printed on it). We can assume that the sheets are infinitely large in any direction. To draw a picture, Leo Jr. colors some of the cells on a sheet gray. He considers the resulting picture beautiful if the following conditions are satisfied: The picture is connected, that is, it is possible to get from any gray cell to any other by following a chain of gray cells, with each pair of adjacent cells in the path being neighbours (that is, sharing a side). Each gray cell has an even number of gray neighbours. There are exactly $n$ gray cells with all gray neighbours. The number of other gray cells can be arbitrary (but reasonable, so that they can all be listed). Leo Jr. is now struggling to draw a beautiful picture with a particular choice of $n$. Help him, and provide any example of a beautiful picture. To output cell coordinates in your answer, assume that the sheet is provided with a Cartesian coordinate system such that one of the cells is chosen to be the origin $(0, 0)$, axes $0x$ and $0y$ are orthogonal and parallel to grid lines, and a unit step along any axis in any direction takes you to a neighbouring cell. -----Input----- The only line contains a single integer $n$ ($1 \leq n \leq 500$) — the number of gray cells with all gray neighbours in a beautiful picture. -----Output----- In the first line, print a single integer $k$ — the number of gray cells in your picture. For technical reasons, $k$ should not exceed $5 \cdot 10^5$. Each of the following $k$ lines should contain two integers — coordinates of a gray cell in your picture. All listed cells should be distinct, and the picture should satisdfy all the properties listed above. All coordinates should not exceed $10^9$ by absolute value. One can show that there exists an answer satisfying all requirements with a small enough $k$. -----Example----- Input 4 Output 12 1 0 2 0 0 1 1 1 2 1 3 1 0 2 1 2 2 2 3 2 1 3 2 3 -----Note----- The answer for the sample is pictured below: [Image] Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. You are given a functional graph. It is a directed graph, in which from each vertex goes exactly one arc. The vertices are numerated from 0 to n - 1. Graph is given as the array f_0, f_1, ..., f_{n} - 1, where f_{i} — the number of vertex to which goes the only arc from the vertex i. Besides you are given array with weights of the arcs w_0, w_1, ..., w_{n} - 1, where w_{i} — the arc weight from i to f_{i}. [Image] The graph from the first sample test. Also you are given the integer k (the length of the path) and you need to find for each vertex two numbers s_{i} and m_{i}, where: s_{i} — the sum of the weights of all arcs of the path with length equals to k which starts from the vertex i; m_{i} — the minimal weight from all arcs on the path with length k which starts from the vertex i. The length of the path is the number of arcs on this path. -----Input----- The first line contains two integers n, k (1 ≤ n ≤ 10^5, 1 ≤ k ≤ 10^10). The second line contains the sequence f_0, f_1, ..., f_{n} - 1 (0 ≤ f_{i} < n) and the third — the sequence w_0, w_1, ..., w_{n} - 1 (0 ≤ w_{i} ≤ 10^8). -----Output----- Print n lines, the pair of integers s_{i}, m_{i} in each line. -----Examples----- Input 7 3 1 2 3 4 3 2 6 6 3 1 4 2 2 3 Output 10 1 8 1 7 1 10 2 8 2 7 1 9 3 Input 4 4 0 1 2 3 0 1 2 3 Output 0 0 4 1 8 2 12 3 Input 5 3 1 2 3 4 0 4 1 2 14 3 Output 7 1 17 1 19 2 21 3 8 1 Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. Okabe needs to renovate the Future Gadget Laboratory after he tried doing some crazy experiments! The lab is represented as an n by n square grid of integers. A good lab is defined as a lab in which every number not equal to 1 can be expressed as the sum of a number in the same row and a number in the same column. In other words, for every x, y such that 1 ≤ x, y ≤ n and a_{x}, y ≠ 1, there should exist two indices s and t so that a_{x}, y = a_{x}, s + a_{t}, y, where a_{i}, j denotes the integer in i-th row and j-th column. Help Okabe determine whether a given lab is good! -----Input----- The first line of input contains the integer n (1 ≤ n ≤ 50) — the size of the lab. The next n lines contain n space-separated integers denoting a row of the grid. The j-th integer in the i-th row is a_{i}, j (1 ≤ a_{i}, j ≤ 10^5). -----Output----- Print "Yes" if the given lab is good and "No" otherwise. You can output each letter in upper or lower case. -----Examples----- Input 3 1 1 2 2 3 1 6 4 1 Output Yes Input 3 1 5 2 1 1 1 1 2 3 Output No -----Note----- In the first sample test, the 6 in the bottom left corner is valid because it is the sum of the 2 above it and the 4 on the right. The same holds for every number not equal to 1 in this table, so the answer is "Yes". In the second sample test, the 5 cannot be formed as the sum of an integer in the same row and an integer in the same column. Thus the answer is "No". Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. There are $n$ children, who study at the school №41. It is well-known that they are good mathematicians. Once at a break, they arranged a challenge for themselves. All children arranged in a row and turned heads either to the left or to the right. Children can do the following: in one second several pairs of neighboring children who are looking at each other can simultaneously turn the head in the opposite direction. For instance, the one who was looking at the right neighbor turns left and vice versa for the second child. Moreover, every second at least one pair of neighboring children performs such action. They are going to finish when there is no pair of neighboring children who are looking at each other. You are given the number $n$, the initial arrangement of children and the number $k$. You have to find a way for the children to act if they want to finish the process in exactly $k$ seconds. More formally, for each of the $k$ moves, you need to output the numbers of the children who turn left during this move. For instance, for the configuration shown below and $k = 2$ children can do the following steps: [Image] At the beginning, two pairs make move: $(1, 2)$ and $(3, 4)$. After that, we receive the following configuration: [Image] At the second move pair $(2, 3)$ makes the move. The final configuration is reached. Good job. [Image] It is guaranteed that if the solution exists, it takes not more than $n^2$ "headturns". -----Input----- The first line of input contains two integers $n$ and $k$ ($2 \le n \le 3000$, $1 \le k \le 3000000$) — the number of children and required number of moves. The next line contains a string of length $n$ and consists only of characters L and R, where L means that the child looks to the left and R means that the child looks to the right. -----Output----- If there is no solution, print a single line with number $-1$. Otherwise, output $k$ lines. Each line has to start with a number $n_i$ ($1\le n_i \le \frac{n}{2}$) — the number of pairs of children, who turn at this move. After that print $n_i$ distinct integers — the numbers of the children who will turn left during this move. After performing all "headturns", there can't be a pair of two neighboring children looking at each other. If there are many solutions, print any of them. -----Examples----- Input 2 1 RL Output 1 1 Input 2 1 LR Output -1 Input 4 2 RLRL Output 2 1 3 1 2 -----Note----- The first sample contains a pair of children who look at each other. After one move, they can finish the process. In the second sample, children can't make any move. As a result, they can't end in $k>0$ moves. The third configuration is described in the statement. Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. You are given an undirected connected weighted graph consisting of $n$ vertices and $m$ edges. Let's denote the length of the shortest path from vertex $1$ to vertex $i$ as $d_i$. You have to erase some edges of the graph so that at most $k$ edges remain. Let's call a vertex $i$ good if there still exists a path from $1$ to $i$ with length $d_i$ after erasing the edges. Your goal is to erase the edges in such a way that the number of good vertices is maximized. -----Input----- The first line contains three integers $n$, $m$ and $k$ ($2 \le n \le 3 \cdot 10^5$, $1 \le m \le 3 \cdot 10^5$, $n - 1 \le m$, $0 \le k \le m$) — the number of vertices and edges in the graph, and the maximum number of edges that can be retained in the graph, respectively. Then $m$ lines follow, each containing three integers $x$, $y$, $w$ ($1 \le x, y \le n$, $x \ne y$, $1 \le w \le 10^9$), denoting an edge connecting vertices $x$ and $y$ and having weight $w$. The given graph is connected (any vertex can be reached from any other vertex) and simple (there are no self-loops, and for each unordered pair of vertices there exists at most one edge connecting these vertices). -----Output----- In the first line print $e$ — the number of edges that should remain in the graph ($0 \le e \le k$). In the second line print $e$ distinct integers from $1$ to $m$ — the indices of edges that should remain in the graph. Edges are numbered in the same order they are given in the input. The number of good vertices should be as large as possible. -----Examples----- Input 3 3 2 1 2 1 3 2 1 1 3 3 Output 2 1 2 Input 4 5 2 4 1 8 2 4 1 2 1 3 3 4 9 3 1 5 Output 2 3 2 Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. You are given an array $a_1, a_2, \dots, a_n$ of integer numbers. Your task is to divide the array into the maximum number of segments in such a way that: each element is contained in exactly one segment; each segment contains at least one element; there doesn't exist a non-empty subset of segments such that bitwise XOR of the numbers from them is equal to $0$. Print the maximum number of segments the array can be divided into. Print -1 if no suitable division exists. -----Input----- The first line contains a single integer $n$ ($1 \le n \le 2 \cdot 10^5$) — the size of the array. The second line contains $n$ integers $a_1, a_2, \dots, a_n$ ($0 \le a_i \le 10^9$). -----Output----- Print the maximum number of segments the array can be divided into while following the given constraints. Print -1 if no suitable division exists. -----Examples----- Input 4 5 5 7 2 Output 2 Input 3 1 2 3 Output -1 Input 3 3 1 10 Output 3 -----Note----- In the first example $2$ is the maximum number. If you divide the array into $\{[5], [5, 7, 2]\}$, the XOR value of the subset of only the second segment is $5 \oplus 7 \oplus 2 = 0$. $\{[5, 5], [7, 2]\}$ has the value of the subset of only the first segment being $5 \oplus 5 = 0$. However, $\{[5, 5, 7], [2]\}$ will lead to subsets $\{[5, 5, 7]\}$ of XOR $7$, $\{[2]\}$ of XOR $2$ and $\{[5, 5, 7], [2]\}$ of XOR $5 \oplus 5 \oplus 7 \oplus 2 = 5$. Let's take a look at some division on $3$ segments — $\{[5], [5, 7], [2]\}$. It will produce subsets: $\{[5]\}$, XOR $5$; $\{[5, 7]\}$, XOR $2$; $\{[5], [5, 7]\}$, XOR $7$; $\{[2]\}$, XOR $2$; $\{[5], [2]\}$, XOR $7$; $\{[5, 7], [2]\}$, XOR $0$; $\{[5], [5, 7], [2]\}$, XOR $5$; As you can see, subset $\{[5, 7], [2]\}$ has its XOR equal to $0$, which is unacceptable. You can check that for other divisions of size $3$ or $4$, non-empty subset with $0$ XOR always exists. The second example has no suitable divisions. The third example array can be divided into $\{[3], [1], [10]\}$. No subset of these segments has its XOR equal to $0$. Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. Vasya is an administrator of a public page of organization "Mouse and keyboard" and his everyday duty is to publish news from the world of competitive programming. For each news he also creates a list of hashtags to make searching for a particular topic more comfortable. For the purpose of this problem we define hashtag as a string consisting of lowercase English letters and exactly one symbol '#' located at the beginning of the string. The length of the hashtag is defined as the number of symbols in it without the symbol '#'. The head administrator of the page told Vasya that hashtags should go in lexicographical order (take a look at the notes section for the definition). Vasya is lazy so he doesn't want to actually change the order of hashtags in already published news. Instead, he decided to delete some suffixes (consecutive characters at the end of the string) of some of the hashtags. He is allowed to delete any number of characters, even the whole string except for the symbol '#'. Vasya wants to pick such a way to delete suffixes that the total number of deleted symbols is minimum possible. If there are several optimal solutions, he is fine with any of them. -----Input----- The first line of the input contains a single integer n (1 ≤ n ≤ 500 000) — the number of hashtags being edited now. Each of the next n lines contains exactly one hashtag of positive length. It is guaranteed that the total length of all hashtags (i.e. the total length of the string except for characters '#') won't exceed 500 000. -----Output----- Print the resulting hashtags in any of the optimal solutions. -----Examples----- Input 3 #book #bigtown #big Output #b #big #big Input 3 #book #cool #cold Output #book #co #cold Input 4 #car #cart #art #at Output # # #art #at Input 3 #apple #apple #fruit Output #apple #apple #fruit -----Note----- Word a_1, a_2, ..., a_{m} of length m is lexicographically not greater than word b_1, b_2, ..., b_{k} of length k, if one of two conditions hold: at first position i, such that a_{i} ≠ b_{i}, the character a_{i} goes earlier in the alphabet than character b_{i}, i.e. a has smaller character than b in the first position where they differ; if there is no such position i and m ≤ k, i.e. the first word is a prefix of the second or two words are equal. The sequence of words is said to be sorted in lexicographical order if each word (except the last one) is lexicographically not greater than the next word. For the words consisting of lowercase English letters the lexicographical order coincides with the alphabet word order in the dictionary. According to the above definition, if a hashtag consisting of one character '#' it is lexicographically not greater than any other valid hashtag. That's why in the third sample we can't keep first two hashtags unchanged and shorten the other two. Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. You are given an integer L. Construct a directed graph that satisfies the conditions below. The graph may contain multiple edges between the same pair of vertices. It can be proved that such a graph always exists. - The number of vertices, N, is at most 20. The vertices are given ID numbers from 1 to N. - The number of edges, M, is at most 60. Each edge has an integer length between 0 and 10^6 (inclusive). - Every edge is directed from the vertex with the smaller ID to the vertex with the larger ID. That is, 1,2,...,N is one possible topological order of the vertices. - There are exactly L different paths from Vertex 1 to Vertex N. The lengths of these paths are all different, and they are integers between 0 and L-1. Here, the length of a path is the sum of the lengths of the edges contained in that path, and two paths are considered different when the sets of the edges contained in those paths are different. -----Constraints----- - 2 \leq L \leq 10^6 - L is an integer. -----Input----- Input is given from Standard Input in the following format: L -----Output----- In the first line, print N and M, the number of the vertices and edges in your graph. In the i-th of the following M lines, print three integers u_i,v_i and w_i, representing the starting vertex, the ending vertex and the length of the i-th edge. If there are multiple solutions, any of them will be accepted. -----Sample Input----- 4 -----Sample Output----- 8 10 1 2 0 2 3 0 3 4 0 1 5 0 2 6 0 3 7 0 4 8 0 5 6 1 6 7 1 7 8 1 In the graph represented by the sample output, there are four paths from Vertex 1 to N=8: - 1 → 2 → 3 → 4 → 8 with length 0 - 1 → 2 → 3 → 7 → 8 with length 1 - 1 → 2 → 6 → 7 → 8 with length 2 - 1 → 5 → 6 → 7 → 8 with length 3 There are other possible solutions. Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. Snuke has decided to play with a six-sided die. Each of its six sides shows an integer 1 through 6, and two numbers on opposite sides always add up to 7. Snuke will first put the die on the table with an arbitrary side facing upward, then repeatedly perform the following operation: - Operation: Rotate the die 90° toward one of the following directions: left, right, front (the die will come closer) and back (the die will go farther). Then, obtain y points where y is the number written in the side facing upward. For example, let us consider the situation where the side showing 1 faces upward, the near side shows 5 and the right side shows 4, as illustrated in the figure. If the die is rotated toward the right as shown in the figure, the side showing 3 will face upward. Besides, the side showing 4 will face upward if the die is rotated toward the left, the side showing 2 will face upward if the die is rotated toward the front, and the side showing 5 will face upward if the die is rotated toward the back. Find the minimum number of operation Snuke needs to perform in order to score at least x points in total. -----Constraints----- - 1 ≦ x ≦ 10^{15} - x is an integer. -----Input----- The input is given from Standard Input in the following format: x -----Output----- Print the answer. -----Sample Input----- 7 -----Sample Output----- 2 Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. We have a tree with N vertices numbered 1 to N. The i-th edge in the tree connects Vertex u_i and Vertex v_i, and its length is w_i. Your objective is to paint each vertex in the tree white or black (it is fine to paint all vertices the same color) so that the following condition is satisfied: - For any two vertices painted in the same color, the distance between them is an even number. Find a coloring of the vertices that satisfies the condition and print it. It can be proved that at least one such coloring exists under the constraints of this problem. -----Constraints----- - All values in input are integers. - 1 \leq N \leq 10^5 - 1 \leq u_i < v_i \leq N - 1 \leq w_i \leq 10^9 -----Input----- Input is given from Standard Input in the following format: N u_1 v_1 w_1 u_2 v_2 w_2 . . . u_{N - 1} v_{N - 1} w_{N - 1} -----Output----- Print a coloring of the vertices that satisfies the condition, in N lines. The i-th line should contain 0 if Vertex i is painted white and 1 if it is painted black. If there are multiple colorings that satisfy the condition, any of them will be accepted. -----Sample Input----- 3 1 2 2 2 3 1 -----Sample Output----- 0 0 1 Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.

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