Arkady decided to buy roses for his girlfriend.A flower shop has white, orange and red roses, and the total amount of them is n. Arkady thinks that red roses are not good together with white roses, so he won't buy a bouquet containing both red and white roses. Also, Arkady won't buy a bouquet where all roses have the same color. Arkady wants to buy exactly k roses. For each rose in the shop he knows its beauty and color: the beauty of the i-th rose is bi, and its color is ci ('W' for a white rose, 'O' for an orange rose and 'R' for a red rose). Compute the maximum possible total beauty of a bouquet of k roses satisfying the constraints above or determine that it is not possible to make such a bouquet.

The first line contains two integers n and k (1 ≤ k ≤ n ≤ 200 000) — the number of roses in the show and the number of roses Arkady wants to buy. The second line contains a sequence of integers b1, b2, ..., bn (1 ≤ bi ≤ 10 000), where bi equals the beauty of the i-th rose. The third line contains a string c of length n, consisting of uppercase English letters 'W', 'O' and 'R', where ci denotes the color of the i-th rose: 'W' denotes white, 'O'  — orange, 'R' — red.

Print the maximum possible total beauty of a bouquet of k roses that satisfies the constraints above. If it is not possible to make a single such bouquet, print -1.

NoteIn the first example Arkady wants to buy 3 roses. He can, for example, buy both red roses (their indices are 1 and 2, and their total beauty is 7) and the only orange rose (its index is 3, its beauty is 4). This way the total beauty of the bouquet is 11. In the second example Arkady can not buy a bouquet because all roses have the same color.


Vitya has learned that the answer for The Ultimate Question of Life, the Universe, and Everything is not the integer 54 42, but an increasing integer sequence $$$a_1, \ldots, a_n$$$. In order to not reveal the secret earlier than needed, Vitya encrypted the answer and obtained the sequence $$$b_1, \ldots, b_n$$$ using the following rules:  $$$b_1 = a_1$$$; $$$b_i = a_i \oplus a_{i - 1}$$$ for all $$$i$$$ from 2 to $$$n$$$, where $$$x \oplus y$$$ is the bitwise XOR of $$$x$$$ and $$$y$$$. It is easy to see that the original sequence can be obtained using the rule $$$a_i = b_1 \oplus \ldots \oplus b_i$$$.However, some time later Vitya discovered that the integers $$$b_i$$$ in the cypher got shuffled, and it can happen that when decrypted using the rule mentioned above, it can produce a sequence that is not increasing. In order to save his reputation in the scientific community, Vasya decided to find some permutation of integers $$$b_i$$$ so that the sequence $$$a_i = b_1 \oplus \ldots \oplus b_i$$$ is strictly increasing. Help him find such a permutation or determine that it is impossible.

The first line contains a single integer $$$n$$$ ($$$1 \leq n \leq 10^5$$$). The second line contains $$$n$$$ integers $$$b_1, \ldots, b_n$$$ ($$$1 \leq b_i < 2^{60}$$$).

If there are no valid permutations, print a single line containing "No". Otherwise in the first line print the word "Yes", and in the second line print integers $$$b'_1, \ldots, b'_n$$$ — a valid permutation of integers $$$b_i$$$. The unordered multisets $$$\{b_1, \ldots, b_n\}$$$ and $$$\{b'_1, \ldots, b'_n\}$$$ should be equal, i. e. for each integer $$$x$$$ the number of occurrences of $$$x$$$ in the first multiset should be equal to the number of occurrences of $$$x$$$ in the second multiset. Apart from this, the sequence $$$a_i = b'_1 \oplus \ldots \oplus b'_i$$$ should be strictly increasing. If there are multiple answers, print any of them.

NoteIn the first example no permutation is valid.In the second example the given answer lead to the sequence $$$a_1 = 4$$$, $$$a_2 = 8$$$, $$$a_3 = 15$$$, $$$a_4 = 16$$$, $$$a_5 = 23$$$, $$$a_6 = 42$$$.


There is a straight line colored in white. n black segments are added on it one by one.After each segment is added, determine the number of connected components of black segments (i. e. the number of black segments in the union of the black segments). In particular, if one segment ends in a point x, and another segment starts in the point x, these two segments belong to the same connected component.

The first line contains a single integer n (1 ≤ n ≤ 200 000) — the number of segments. The i-th of the next n lines contains two integers li and ri (1 ≤ li < ri ≤ 109) — the coordinates of the left and the right ends of the i-th segment. The segments are listed in the order they are added on the white line.

Print n integers — the number of connected components of black segments after each segment is added.

NoteIn the first example there are two components after the addition of the first two segments, because these segments do not intersect. The third added segment intersects the left segment and touches the right segment at the point 4 (these segments belong to the same component, according to the statements). Thus the number of connected components of black segments is equal to 1 after that.


Polycarp has a strict daily schedule. He has n alarms set for each day, and the i-th alarm rings each day at the same time during exactly one minute.Determine the longest time segment when Polycarp can sleep, i. e. no alarm rings in that period. It is possible that Polycarp begins to sleep in one day, and wakes up in another.

The first line contains a single integer n (1 ≤ n ≤ 100) — the number of alarms. Each of the next n lines contains a description of one alarm. Each description has a format "hh:mm", where hh is the hour when the alarm rings, and mm is the minute of that hour when the alarm rings. The number of hours is between 0 and 23, and the number of minutes is between 0 and 59. All alarm times are distinct. The order of the alarms is arbitrary. Each alarm starts ringing in the beginning of the corresponding minute and rings for exactly one minute (i. e. stops ringing in the beginning of the next minute). Polycarp can start sleeping instantly when no alarm is ringing, and he wakes up at the moment when some alarm starts ringing.

Print a line in format "hh:mm", denoting the maximum time Polycarp can sleep continuously. hh denotes the number of hours, and mm denotes the number of minutes. The number of minutes should be between 0 and 59. Look through examples to understand the format better.

NoteIn the first example there is only one alarm which rings during one minute of a day, and then rings again on the next day, 23 hours and 59 minutes later. Polycarp can sleep all this time.


There are times you recall a good old friend and everything you've come through together. Luckily there are social networks — they store all your message history making it easy to know what you argued over 10 years ago.More formal, your message history is a sequence of messages ordered by time sent numbered from 1 to n where n is the total number of messages in the chat.Each message might contain a link to an earlier message which it is a reply to. When opening a message x or getting a link to it, the dialogue is shown in such a way that k previous messages, message x and k next messages are visible (with respect to message x). In case there are less than k messages somewhere, they are yet all shown.Digging deep into your message history, you always read all visible messages and then go by the link in the current message x (if there is one) and continue reading in the same manner.Determine the number of messages you'll read if your start from message number t for all t from 1 to n. Calculate these numbers independently. If you start with message x, the initial configuration is x itself, k previous and k next messages. Messages read multiple times are considered as one.

The first line contains two integers n and k (1 ≤ n ≤ 105, 0 ≤ k ≤ n) — the total amount of messages and the number of previous and next messages visible. The second line features a sequence of integers a1, a2, ..., an (0 ≤ ai < i), where ai denotes the i-th message link destination or zero, if there's no link from i. All messages are listed in chronological order. It's guaranteed that the link from message x goes to message with number strictly less than x.

Print n integers with i-th denoting the number of distinct messages you can read starting from message i and traversing the links while possible.

NoteConsider i = 6 in sample case one. You will read message 6, then 2, then 1 and then there will be no link to go.In the second sample case i = 6 gives you messages 5, 6, 7 since k = 1, then 4, 5, 6, then 2, 3, 4 and then the link sequence breaks. The number of distinct messages here is equal to 6.


When registering in a social network, users are allowed to create their own convenient login to make it easier to share contacts, print it on business cards, etc.Login is an arbitrary sequence of lower and uppercase latin letters, digits and underline symbols («_»). However, in order to decrease the number of frauds and user-inattention related issues, it is prohibited to register a login if it is similar with an already existing login. More precisely, two logins s and t are considered similar if we can transform s to t via a sequence of operations of the following types:   transform lowercase letters to uppercase and vice versa;  change letter «O» (uppercase latin letter) to digit «0» and vice versa;  change digit «1» (one) to any letter among «l» (lowercase latin «L»), «I» (uppercase latin «i») and vice versa, or change one of these letters to other. For example, logins «Codeforces» and «codef0rces» as well as «OO0OOO00O0OOO0O00OOO0OO_lol» and «OO0OOO0O00OOO0O00OO0OOO_1oI» are considered similar whereas «Codeforces» and «Code_forces» are not.You're given a list of existing logins with no two similar amonst and a newly created user login. Check whether this new login is similar with any of the existing ones.

The first line contains a non-empty string s consisting of lower and uppercase latin letters, digits and underline symbols («_») with length not exceeding 50  — the login itself. The second line contains a single integer n (1 ≤ n ≤ 1 000) — the number of existing logins. The next n lines describe the existing logins, following the same constraints as the user login (refer to the first line of the input). It's guaranteed that no two existing logins are similar.

Print «Yes» (without quotes), if user can register via this login, i.e. none of the existing logins is similar with it. Otherwise print «No» (without quotes).

NoteIn the second sample case the user wants to create a login consisting of three zeros. It's impossible due to collision with the third among the existing.In the third sample case the new login is similar with the second one.


Arcady is a copywriter. His today's task is to type up an already well-designed story using his favorite text editor.Arcady types words, punctuation signs and spaces one after another. Each letter and each sign (including line feed) requires one keyboard click in order to be printed. Moreover, when Arcady has a non-empty prefix of some word on the screen, the editor proposes a possible autocompletion for this word, more precisely one of the already printed words such that its prefix matches the currently printed prefix if this word is unique. For example, if Arcady has already printed «codeforces», «coding» and «codeforces» once again, then there will be no autocompletion attempt for «cod», but if he proceeds with «code», the editor will propose «codeforces».With a single click Arcady can follow the editor's proposal, i.e. to transform the current prefix to it. Note that no additional symbols are printed after the autocompletion (no spaces, line feeds, etc). What is the minimum number of keyboard clicks Arcady has to perform to print the entire text, if he is not allowed to move the cursor or erase the already printed symbols?A word here is a contiguous sequence of latin letters bordered by spaces, punctuation signs and line/text beginnings/ends. Arcady uses only lowercase letters. For example, there are 20 words in «it's well-known that tic-tac-toe is a paper-and-pencil game for two players, x and o.».

The only line contains Arcady's text, consisting only of lowercase latin letters, spaces, line feeds and the following punctuation signs: «.», «,», «?», «!», «'» and «-». The total amount of symbols doesn't exceed 3·105. It's guaranteed that all lines are non-empty.

Print a single integer — the minimum number of clicks.

NoteIn sample case one it's optimal to use autocompletion for the first instance of «snowboarding» after typing up «sn» and for the second instance of «snowboarding» after typing up «snowb». This will save 7 clicks.In sample case two it doesn't matter whether to use autocompletion or not.

snow affects sports such as skiing, snowboarding, and snowmachine travel.
snowboarding is a recreational activity and olympic and paralympic sport.

thun-thun-thunder, thunder, thunder
thunder, thun-, thunder
thun-thun-thunder, thunder
thunder, feel the thunder
lightning then the thunder
thunder, feel the thunder
lightning then the thunder
thunder, thunder


В самолёте есть n рядов мест. Если смотреть на ряды сверху, то в каждом ряду есть 3 места слева, затем проход между рядами, затем 4 центральных места, затем ещё один проход между рядами, а затем ещё 3 места справа.Известно, что некоторые места уже заняты пассажирами. Всего есть два вида пассажиров — статусные (те, которые часто летают) и обычные. Перед вами стоит задача рассадить ещё k обычных пассажиров так, чтобы суммарное число соседей у статусных пассажиров было минимально возможным. Два пассажира считаются соседями, если они сидят в одном ряду и между ними нет других мест и прохода между рядами. Если пассажир является соседним пассажиром для двух статусных пассажиров, то его следует учитывать в сумме соседей дважды.

стандартный ввод

стандартный вывод

256 мегабайт

В первой строке следуют два целых числа n и k (1 ≤ n ≤ 100, 1 ≤ k ≤ 10·n) — количество рядов мест в самолёте и количество пассажиров, которых нужно рассадить. Далее следует описание рядов мест самолёта по одному ряду в строке. Если очередной символ равен '-', то это проход между рядами. Если очередной символ равен '.', то это свободное место. Если очередной символ равен 'S', то на текущем месте будет сидеть статусный пассажир. Если очередной символ равен 'P', то на текущем месте будет сидеть обычный пассажир.  Гарантируется, что количество свободных мест не меньше k. Гарантируется, что все ряды удовлетворяют описанному в условии формату.

В первую строку выведите минимальное суммарное число соседей у статусных пассажиров. Далее выведите план рассадки пассажиров, который минимизирует суммарное количество соседей у статусных пассажиров, в том же формате, что и во входных данных. Если в свободное место нужно посадить одного из k пассажиров, выведите строчную букву 'x' вместо символа '.'.

ПримечаниеВ первом примере нужно посадить ещё двух обычных пассажиров. Для минимизации соседей у статусных пассажиров, нужно посадить первого из них на третье слева место, а второго на любое из оставшихся двух мест, так как независимо от выбора места он станет соседом двух статусных пассажиров. Изначально, у статусного пассажира, который сидит на самом левом месте уже есть сосед. Также на четвёртом и пятом местах слева сидят статусные пассажиры, являющиеся соседями друг для друга (что добавляет к сумме 2).Таким образом, после посадки ещё двух обычных пассажиров, итоговое суммарное количество соседей у статусных пассажиров станет равно пяти.

4 9
PP.-PPPS-S.S
PSP-PPSP-.S.
.S.-S..P-SS.
P.S-P.PP-PSP

15
PPx-PPPS-S.S
PSP-PPSP-xSx
xSx-SxxP-SSx
P.S-PxPP-PSP


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.

стандартный ввод

стандартный вывод

256 мегабайт

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.

Выведите одно целое число — количество возможных стартовых составов.

ПримечаниеВ 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.


Polycarp is currently developing a project in Vaja language and using a popular dependency management system called Vamen. From Vamen's point of view both Vaja project and libraries are treated projects for simplicity.A project in Vaja has its own uniqie non-empty name consisting of lowercase latin letters with length not exceeding 10 and version — positive integer from 1 to 106. Each project (keep in mind that it is determined by both its name and version) might depend on other projects. For sure, there are no cyclic dependencies.You're given a list of project descriptions. The first of the given projects is the one being developed by Polycarp at this moment. Help Polycarp determine all projects that his project depends on (directly or via a certain chain). It's possible that Polycarp's project depends on two different versions of some project. In this case collision resolving is applied, i.e. for each such project the system chooses the version that minimizes the distance from it to Polycarp's project. If there are several options, the newer (with the maximum version) is preferred. This version is considered actual; other versions and their dependencies are ignored.More formal, choose such a set of projects of minimum possible size that the following conditions hold:   Polycarp's project is chosen;  Polycarp's project depends (directly or indirectly) on all other projects in the set;  no two projects share the name;  for each project x that some other project in the set depends on we have either x or some y with other version and shorter chain to Polycarp's project chosen. In case of ties the newer one is chosen. Output all Polycarp's project's dependencies (Polycarp's project itself should't be printed) in lexicographical order.

The first line contains an only integer n (1 ≤ n ≤ 1 000) — the number of projects in Vaja. The following lines contain the project descriptions. Each project is described by a line consisting of its name and version separated by space. The next line gives the number of direct dependencies (from 0 to n - 1) and the dependencies themselves (one in a line) in arbitrary order. Each dependency is specified by its name and version. The projects are also given in arbitrary order, but the first of them is always Polycarp's. Project descriptions are separated by one empty line. Refer to samples for better understanding. It's guaranteed that there are no cyclic dependencies.

Output all Polycarp's project's dependencies in lexicographical order.

NoteThe first sample is given in the pic below. Arrow from A to B means that B directly depends on A. Projects that Polycarp's project «a» (version 3) depends on are painted black.  The second sample is again given in the pic below. Arrow from A to B means that B directly depends on A. Projects that Polycarp's project «codehorses» (version 5) depends on are paint it black. Note that «extra 1» is chosen instead of «extra 3» since «mashadb 1» and all of its dependencies are ignored due to «mashadb 2».

9
codehorses 5
3
webfrmk 6
mashadb 1
mashadb 2

commons 2
0

mashadb 3
0

webfrmk 6
2
mashadb 3
commons 2

extra 4
1
extra 3

extra 3
0

extra 1
0

mashadb 1
1
extra 3

mashadb 2
1
extra 1


Герой Аркадий находится на узкой полоске земли, разделенной на n зон, пронумерованных от 1 до n. Из i-й зоны можно пройти лишь в (i - 1)-ю зону и в (i + 1)-ю зону, если они существуют. При этом между каждой парой соседних зон находятся пограничные врата, которые могут быть разных цветов, цвет врат между i-й и (i + 1)-й зоной равен gi.Аркадий может пройти пограничные врата некоторого цвета, только если он перед этим побывал в одном из шатров хранителей ключей этого цвета и взял ключ. В каждой зоне находится ровно один шатер хранителя ключей некоторого цвета, цвет шатра в i-й зоне равен ki. После посещения шатра определенного цвета Аркадий может неограниченное число раз проходить через любые врата этого цвета.На проход через одни врата Аркадий тратит один ход, на посещение шатра и другие перемещения ходы не требуются. За какое минимальное число ходов Аркадий может попасть из зоны a в зону b, если изначально у него нет никаких ключей?

стандартный ввод

стандартный вывод

256 мегабайт

Первая строка содержит три целых числа n, a, b (2 ≤ n ≤ 100 000, 1 ≤ a, b ≤ n, a ≠ b) — число зон, номер начальной зоны и номер конечной зоны, соответственно. Вторая строка содержит n - 1 целое число g1, g2, ..., gn - 1 (1 ≤ gi ≤ 100 000), где gi означает цвет пограничных врат между зонами i и i + 1. Третья строка содержит n целых чисел k1, k2, ..., kn (1 ≤ ki ≤ 100 000), где ki означает цвет шатра хранителя ключей в i-й зоне.

Если Аркадий не может попасть из зоны a в зону b, не имея изначально ключей, выведите -1. Иначе выведите минимальное количество ходов, которое потребуется Аркадию.

ПримечаниеВ первом примере, чтобы попасть из зоны 4 в зону 1, Аркадию нужно сначала взять ключ цвета 1, пройти в зону 3, там взять ключ цвета 2 и пройти обратно в зону 4 и затем в зону 5, взять там ключ цвета 3 и дойти до зоны 1 за четыре хода.Во втором примере Аркадий может дойти лишь до четвертой зоны, так как шатров хранителей ключей цвета 1 нет совсем.


Как известно, в теплую погоду многие жители крупных городов пользуются сервисами городского велопроката. Вот и Аркадий сегодня будет добираться от школы до дома, используя городские велосипеды.Школа и дом находятся на одной прямой улице, кроме того, на той же улице есть n точек, где можно взять велосипед в прокат или сдать его. Первый велопрокат находится в точке x1 километров вдоль улицы, второй — в точке x2 и так далее, n-й велопрокат находится в точке xn. Школа Аркадия находится в точке x1 (то есть там же, где и первый велопрокат), а дом — в точке xn (то есть там же, где и n-й велопрокат). Известно, что xi < xi + 1 для всех 1 ≤ i < n.Согласно правилам пользования велопроката, Аркадий может брать велосипед в прокат только на ограниченное время, после этого он должен обязательно вернуть его в одной из точек велопроката, однако, он тут же может взять новый велосипед, и отсчет времени пойдет заново. Аркадий может брать не более одного велосипеда в прокат одновременно. Если Аркадий решает взять велосипед в какой-то точке проката, то он сдаёт тот велосипед, на котором он до него доехал, берёт ровно один новый велосипед и продолжает на нём своё движение.За отведенное время, независимо от выбранного велосипеда, Аркадий успевает проехать не больше k километров вдоль улицы. Определите, сможет ли Аркадий доехать на велосипедах от школы до дома, и если да, то какое минимальное число раз ему необходимо будет взять велосипед в прокат, включая первый велосипед? Учтите, что Аркадий не намерен сегодня ходить пешком.

стандартный ввод

стандартный вывод

256 мегабайт

В первой строке следуют два целых числа n и k (2 ≤ n ≤ 1 000, 1 ≤ k ≤ 100 000) — количество велопрокатов и максимальное расстояние, которое Аркадий может проехать на одном велосипеде. В следующей строке следует последовательность целых чисел x1, x2, ..., xn (0 ≤ x1 < x2 < ... < xn ≤ 100 000) — координаты точек, в которых находятся велопрокаты. Гарантируется, что координаты велопрокатов заданы в порядке возрастания.

Если Аркадий не сможет добраться от школы до дома только на велосипедах, выведите -1. В противном случае, выведите минимальное количество велосипедов, которые Аркадию нужно взять в точках проката.

ПримечаниеВ первом примере Аркадий должен взять первый велосипед в первом велопрокате и доехать на нём до второго велопроката. Во втором велопрокате он должен взять новый велосипед, на котором он сможет добраться до четвертого велопроката, рядом с которым и находится его дом. Поэтому Аркадию нужно всего два велосипеда, чтобы добраться от школы до дома.Во втором примере всего два велопроката, расстояние между которыми 10. Но максимальное расстояние, которое можно проехать на одном велосипеде, равно 9. Поэтому Аркадий не сможет добраться от школы до дома только на велосипедах.


In Arcady's garden there grows a peculiar apple-tree that fruits one time per year. Its peculiarity can be explained in following way: there are n inflorescences, numbered from 1 to n. Inflorescence number 1 is situated near base of tree and any other inflorescence with number i (i > 1) is situated at the top of branch, which bottom is pi-th inflorescence and pi < i.Once tree starts fruiting, there appears exactly one apple in each inflorescence. The same moment as apples appear, they start to roll down along branches to the very base of tree. Each second all apples, except ones in first inflorescence simultaneously roll down one branch closer to tree base, e.g. apple in a-th inflorescence gets to pa-th inflorescence. Apples that end up in first inflorescence are gathered by Arcady in exactly the same moment. Second peculiarity of this tree is that once two apples are in same inflorescence they annihilate. This happens with each pair of apples, e.g. if there are 5 apples in same inflorescence in same time, only one will not be annihilated and if there are 8 apples, all apples will be annihilated. Thus, there can be no more than one apple in each inflorescence in each moment of time.Help Arcady with counting number of apples he will be able to collect from first inflorescence during one harvest.

First line of input contains single integer number n (2 ≤ n ≤ 100 000)  — number of inflorescences. Second line of input contains sequence of n - 1 integer numbers p2, p3, ..., pn (1 ≤ pi < i), where pi is number of inflorescence into which the apple from i-th inflorescence rolls down.

Single line of output should contain one integer number: amount of apples that Arcady will be able to collect from first inflorescence during one harvest.

NoteIn first example Arcady will be able to collect only one apple, initially situated in 1st inflorescence. In next second apples from 2nd and 3rd inflorescences will roll down and annihilate, and Arcady won't be able to collect them.In the second example Arcady will be able to collect 3 apples. First one is one initially situated in first inflorescence. In a second apple from 2nd inflorescence will roll down to 1st (Arcady will collect it) and apples from 3rd, 4th, 5th inflorescences will roll down to 2nd. Two of them will annihilate and one not annihilated will roll down from 2-nd inflorescence to 1st one in the next second and Arcady will collect it.


Vasya and Kolya play a game with a string, using the following rules. Initially, Kolya creates a string s, consisting of small English letters, and uniformly at random chooses an integer k from a segment [0, len(s) - 1]. He tells Vasya this string s, and then shifts it k letters to the left, i. e. creates a new string t = sk + 1sk + 2... sns1s2... sk. Vasya does not know the integer k nor the string t, but he wants to guess the integer k. To do this, he asks Kolya to tell him the first letter of the new string, and then, after he sees it, open one more letter on some position, which Vasya can choose.Vasya understands, that he can't guarantee that he will win, but he wants to know the probability of winning, if he plays optimally. He wants you to compute this probability. Note that Vasya wants to know the value of k uniquely, it means, that if there are at least two cyclic shifts of s that fit the information Vasya knowns, Vasya loses. Of course, at any moment of the game Vasya wants to maximize the probability of his win.

The only string contains the string s of length l (3 ≤ l ≤ 5000), consisting of small English letters only.

Print the only number — the answer for the problem. You answer is considered correct, if its absolute or relative error does not exceed 10 - 6. Formally, let your answer be a, and the jury's answer be b. Your answer is considered correct if

NoteIn the first example Vasya can always open the second letter after opening the first letter, and the cyclic shift is always determined uniquely.In the second example if the first opened letter of t is "t" or "c", then Vasya can't guess the shift by opening only one other letter. On the other hand, if the first letter is "i" or "a", then he can open the fourth letter and determine the shift uniquely.


Young Teodor enjoys drawing. His favourite hobby is drawing segments with integer borders inside his huge [1;m] segment. One day Teodor noticed that picture he just drawn has one interesting feature: there doesn't exist an integer point, that belongs each of segments in the picture. Having discovered this fact, Teodor decided to share it with Sasha.Sasha knows that Teodor likes to show off so he never trusts him. Teodor wants to prove that he can be trusted sometimes, so he decided to convince Sasha that there is no such integer point in his picture, which belongs to each segment. However Teodor is lazy person and neither wills to tell Sasha all coordinates of segments' ends nor wills to tell him their amount, so he suggested Sasha to ask him series of questions 'Given the integer point xi, how many segments in Fedya's picture contain that point?', promising to tell correct answers for this questions.Both boys are very busy studying and don't have much time, so they ask you to find out how many questions can Sasha ask Teodor, that having only answers on his questions, Sasha can't be sure that Teodor isn't lying to him. Note that Sasha doesn't know amount of segments in Teodor's picture. Sure, Sasha is smart person and never asks about same point twice.

First line of input contains two integer numbers: n and m (1 ≤ n, m ≤ 100 000) — amount of segments of Teodor's picture and maximal coordinate of point that Sasha can ask about. ith of next n lines contains two integer numbers li and ri (1 ≤ li ≤ ri ≤ m) — left and right ends of ith segment in the picture. Note that that left and right ends of segment can be the same point. It is guaranteed that there is no integer point, that belongs to all segments.

Single line of output should contain one integer number k – size of largest set (xi, cnt(xi)) where all xi are different, 1 ≤ xi ≤ m, and cnt(xi) is amount of segments, containing point with coordinate xi, such that one can't be sure that there doesn't exist point, belonging to all of segments in initial picture, if he knows only this set(and doesn't know n).

NoteFirst example shows situation where Sasha can never be sure that Teodor isn't lying to him, because even if one knows cnt(xi) for each point in segment [1;4], he can't distinguish this case from situation Teodor has drawn whole [1;4] segment.In second example Sasha can ask about 5 points e.g. 1, 2, 3, 5, 6, still not being sure if Teodor haven't lied to him. But once he knows information about all points in [1;6] segment, Sasha can be sure that Teodor haven't lied to him.


Arkady and Kirill visited an exhibition of rare coins. The coins were located in a row and enumerated from left to right from 1 to k, each coin either was laid with its obverse (front) side up, or with its reverse (back) side up.Arkady and Kirill made some photos of the coins, each photo contained a segment of neighboring coins. Akrady is interested in obverses, so on each photo made by him there is at least one coin with obverse side up. On the contrary, Kirill is interested in reverses, so on each photo made by him there is at least one coin with its reverse side up.The photos are lost now, but Arkady and Kirill still remember the bounds of the segments of coins each photo contained. Given this information, compute the remainder of division by 109 + 7 of the number of ways to choose the upper side of each coin in such a way, that on each Arkady's photo there is at least one coin with obverse side up, and on each Kirill's photo there is at least one coin with reverse side up.

The first line contains three integers k, n and m (1 ≤ k ≤ 109, 0 ≤ n, m ≤ 105) — the total number of coins, the number of photos made by Arkady, and the number of photos made by Kirill, respectively. The next n lines contain the descriptions of Arkady's photos, one per line. Each of these lines contains two integers l and r (1 ≤ l ≤ r ≤ k), meaning that among coins from the l-th to the r-th there should be at least one with obverse side up. The next m lines contain the descriptions of Kirill's photos, one per line. Each of these lines contains two integers l and r (1 ≤ l ≤ r ≤ k), meaning that among coins from the l-th to the r-th there should be at least one with reverse side up.

Print the only line — the number of ways to choose the side for each coin modulo 109 + 7 = 1000000007.

NoteIn the first example the following ways are possible ('O' — obverse, 'R' — reverse side):   OROOR,  ORORO,  ORORR,  RROOR,  RRORO,  RRORR,  ORROR,  ORRRO. In the second example the information is contradictory: the second coin should have obverse and reverse sides up at the same time, that is impossible. So, the answer is 0.


Two friends are on the coordinate axis Ox in points with integer coordinates. One of them is in the point x1 = a, another one is in the point x2 = b. Each of the friends can move by one along the line in any direction unlimited number of times. When a friend moves, the tiredness of a friend changes according to the following rules: the first move increases the tiredness by 1, the second move increases the tiredness by 2, the third — by 3 and so on. For example, if a friend moves first to the left, then to the right (returning to the same point), and then again to the left his tiredness becomes equal to 1 + 2 + 3 = 6.The friends want to meet in a integer point. Determine the minimum total tiredness they should gain, if they meet in the same point.

The first line contains a single integer a (1 ≤ a ≤ 1000) — the initial position of the first friend.  The second line contains a single integer b (1 ≤ b ≤ 1000) — the initial position of the second friend. It is guaranteed that a ≠ b.

Print the minimum possible total tiredness if the friends meet in the same point.

NoteIn the first example the first friend should move by one to the right (then the meeting happens at point 4), or the second friend should move by one to the left (then the meeting happens at point 3). In both cases, the total tiredness becomes 1.In the second example the first friend should move by one to the left, and the second friend should move by one to the right. Then they meet in the point 100, and the total tiredness becomes 1 + 1 = 2.In the third example one of the optimal ways is the following. The first friend should move three times to the right, and the second friend — two times to the left. Thus the friends meet in the point 8, and the total tiredness becomes 1 + 2 + 3 + 1 + 2 = 9.


The last stage of Football World Cup is played using the play-off system.There are n teams left in this stage, they are enumerated from 1 to n. Several rounds are held, in each round the remaining teams are sorted in the order of their ids, then the first in this order plays with the second, the third — with the fourth, the fifth — with the sixth, and so on. It is guaranteed that in each round there is even number of teams. The winner of each game advances to the next round, the loser is eliminated from the tournament, there are no draws. In the last round there is the only game with two remaining teams: the round is called the Final, the winner is called the champion, and the tournament is over.Arkady wants his two favorite teams to play in the Final. Unfortunately, the team ids are already determined, and it may happen that it is impossible for teams to meet in the Final, because they are to meet in some earlier stage, if they are strong enough. Determine, in which round the teams with ids a and b can meet.

The only line contains three integers n, a and b (2 ≤ n ≤ 256, 1 ≤ a, b ≤ n) — the total number of teams, and the ids of the teams that Arkady is interested in.  It is guaranteed that n is such that in each round an even number of team advance, and that a and b are not equal.

In the only line print "Final!" (without quotes), if teams a and b can meet in the Final. Otherwise, print a single integer — the number of the round in which teams a and b can meet. The round are enumerated from 1.

NoteIn the first example teams 1 and 2 meet in the first round.In the second example teams 2 and 6 can only meet in the third round, which is the Final, if they win all their opponents in earlier rounds.In the third example the teams with ids 7 and 5 can meet in the second round, if they win their opponents in the first round.


Consider the following game for two players. There is one white token and some number of black tokens. Each token is placed on a plane in a point with integer coordinates x and y.The players take turn making moves, white starts. On each turn, a player moves all tokens of their color by 1 to up, down, left or right. Black player can choose directions for each token independently.After a turn of the white player the white token can not be in a point where a black token is located. There are no other constraints on locations of the tokens: positions of black tokens can coincide, after a turn of the black player and initially the white token can be in the same point with some black point. If at some moment the white player can't make a move, he loses. If the white player makes 10100500 moves, he wins.You are to solve the following problem. You are given initial positions of all black tokens. It is guaranteed that initially all these positions are distinct. In how many places can the white token be located initially so that if both players play optimally, the black player wins?

The first line contains a single integer n (1 ≤ n ≤ 105) — the number of black points. The (i + 1)-th line contains two integers xi, yi ( - 105 ≤ xi, yi,  ≤ 105) — the coordinates of the point where the i-th black token is initially located. It is guaranteed that initial positions of black tokens are distinct.

Print the number of points where the white token can be located initially, such that if both players play optimally, the black player wins.

NoteIn the first and second examples initial positions of black tokens are shown with black points, possible positions of the white token (such that the black player wins) are shown with white points.The first example: The second example: In the third example the white tokens should be located in the inner square 2 × 2, to make the black player win.


Anya and Kirill are doing a physics laboratory work. In one of the tasks they have to measure some value n times, and then compute the average value to lower the error.Kirill has already made his measurements, and has got the following integer values: x1, x2, ..., xn. It is important that the values are close to each other, namely, the difference between the maximum value and the minimum value is at most 2.Anya does not want to make the measurements, however, she can't just copy the values from Kirill's work, because the error of each measurement is a random value, and this coincidence will be noted by the teacher. Anya wants to write such integer values y1, y2, ..., yn in her work, that the following conditions are met:  the average value of x1, x2, ..., xn is equal to the average value of y1, y2, ..., yn; all Anya's measurements are in the same bounds as all Kirill's measurements, that is, the maximum value among Anya's values is not greater than the maximum value among Kirill's values, and the minimum value among Anya's values is not less than the minimum value among Kirill's values; the number of equal measurements in Anya's work and Kirill's work is as small as possible among options with the previous conditions met. Formally, the teacher goes through all Anya's values one by one, if there is equal value in Kirill's work and it is not strike off yet, he strikes off this Anya's value and one of equal values in Kirill's work. The number of equal measurements is then the total number of strike off values in Anya's work. Help Anya to write such a set of measurements that the conditions above are met.

The first line contains a single integer n (1 ≤ n ≤ 100 000) — the numeber of measurements made by Kirill. The second line contains a sequence of integers x1, x2, ..., xn ( - 100 000 ≤ xi ≤ 100 000) — the measurements made by Kirill. It is guaranteed that the difference between the maximum and minimum values among values x1, x2, ..., xn does not exceed 2.

In the first line print the minimum possible number of equal measurements. In the second line print n integers y1, y2, ..., yn — the values Anya should write. You can print the integers in arbitrary order. Keep in mind that the minimum value among Anya's values should be not less that the minimum among Kirill's values, and the maximum among Anya's values should be not greater than the maximum among Kirill's values. If there are multiple answers, print any of them.

NoteIn the first example Anya can write zeros as here measurements results. The average value is then equal to the average value of Kirill's values, and there are only two equal measurements.In the second example Anya should write two values 100 and one value 101 (in any order), because it is the only possibility to make the average be the equal to the average of Kirill's values. Thus, all three measurements are equal.In the third example the number of equal measurements is 5.


Let us define two functions f and g on positive integer numbers.   You need to process Q queries. In each query, you will be given three integers l, r and k. You need to print the number of integers x between l and r inclusive, such that g(x) = k.

The first line of the input contains an integer Q (1 ≤ Q ≤ 2 × 105) representing the number of queries.  Q lines follow, each of which contains 3 integers l, r and k (1 ≤ l ≤ r ≤ 106, 1 ≤ k ≤ 9).

For each query, print a single line containing the answer for that query.

NoteIn the first example:  g(33) = 9 as g(33) = g(3 × 3) = g(9) = 9  g(47) = g(48) = g(60) = g(61) = 6  There are no such integers between 47 and 55.  g(4) = g(14) = g(22) = g(27) = g(39) = g(40) = g(41) = g(58) = 4


You are given a string A. Find a string B, where B is a palindrome and A is a subsequence of B.A subsequence of a string is a string that can be derived from it by deleting some (not necessarily consecutive) characters without changing the order of the remaining characters. For example, "cotst" is a subsequence of "contest".A palindrome is a string that reads the same forward or backward.The length of string B should be at most 104. It is guaranteed that there always exists such string.You do not need to find the shortest answer, the only restriction is that the length of string B should not exceed 104.

First line contains a string A (1 ≤ |A| ≤ 103) consisting of lowercase Latin letters, where |A| is a length of A.

Output single line containing B consisting of only lowercase Latin letters. You do not need to find the shortest answer, the only restriction is that the length of string B should not exceed 104. If there are many possible B, print any of them.

NoteIn the first example, "aba" is a subsequence of "aba" which is a palindrome.In the second example, "ab" is a subsequence of "aabaa" which is a palindrome.


For a permutation P[1... N] of integers from 1 to N, function f is defined as follows:  Let g(i) be the minimum positive integer j such that f(i, j) = i. We can show such j always exists.For given N, A, B, find a permutation P of integers from 1 to N such that for 1 ≤ i ≤ N, g(i) equals either A or B.

The only line contains three integers N, A, B (1 ≤ N ≤ 106, 1 ≤ A, B ≤ N).

If no such permutation exists, output -1. Otherwise, output a permutation of integers from 1 to N.

NoteIn the first example, g(1) = g(6) = g(7) = g(9) = 2 and g(2) = g(3) = g(4) = g(5) = g(8) = 5 In the second example, g(1) = g(2) = g(3) = 1


You are given a node of the tree with index 1 and with weight 0. Let cnt be the number of nodes in the tree at any instant (initially, cnt is set to 1). Support Q queries of following two types:   Add a new node (index cnt + 1) with weight W and add edge between node R and this node.   Output the maximum length of sequence of nodes which   starts with R.  Every node in the sequence is an ancestor of its predecessor.  Sum of weight of nodes in sequence does not exceed X.  For some nodes i, j that are consecutive in the sequence if i is an ancestor of j then w[i] ≥ w[j] and there should not exist a node k on simple path from i to j such that w[k] ≥ w[j]  The tree is rooted at node 1 at any instant.Note that the queries are given in a modified way.

First line containing the number of queries Q (1 ≤ Q ≤ 400000). Let last be the answer for previous query of type 2 (initially last equals 0). Each of the next Q lines contains a query of following form:    1 p q (1 ≤ p, q ≤ 1018): This is query of first type where  and . It is guaranteed that 1 ≤ R ≤ cnt and 0 ≤ W ≤ 109.  2 p q (1 ≤ p, q ≤ 1018): This is query of second type where  and . It is guaranteed that 1 ≤ R ≤ cnt and 0 ≤ X ≤ 1015.   denotes bitwise XOR of a and b. It is guaranteed that at least one query of type 2 exists.

Output the answer to each query of second type in separate line.

NoteIn the first example, last = 0- Query 1: 1 1 1, Node 2 with weight 1 is added to node 1.- Query 2: 2 2 0, No sequence of nodes starting at 2 has weight less than or equal to 0. last = 0 - Query 3: 2 2 1, Answer is 1 as sequence will be {2}. last = 1- Query 4: 1 2 1, Node 3 with weight 1 is added to node 2. - Query 5: 2 3 1, Answer is 1 as sequence will be {3}. Node 2 cannot be added as sum of weights cannot be greater than 1. last = 1- Query 6: 2 3 3, Answer is 2 as sequence will be {3, 2}. last = 2


You are given a tree with n nodes (numbered from 1 to n) rooted at node 1. Also, each node has two values associated with it. The values for i-th node are ai and bi.You can jump from a node to any node in its subtree. The cost of one jump from node x to node y is the product of ax and by. The total cost of a path formed by one or more jumps is sum of costs of individual jumps. For every node, calculate the minimum total cost to reach any leaf from that node. Pay attention, that root can never be leaf, even if it has degree 1.Note that you cannot jump from a node to itself.

The first line of input contains an integer n (2 ≤ n ≤ 105) — the number of nodes in the tree. The second line contains n space-separated integers a1,  a2,  ...,  an( - 105  ≤  ai  ≤  105). The third line contains n space-separated integers b1,  b2,  ...,  bn( - 105  ≤  bi  ≤  105). Next n  -  1 lines contains two space-separated integers ui and vi (1 ≤ ui,  vi ≤  n) describing edge between nodes ui and vi in the tree.

Output n space-separated integers, i-th of which denotes the minimum cost of a path from node i to reach any leaf.

NoteIn the first example, node 3 is already a leaf, so the cost is 0. For node 2, jump to node 3 with cost a2 × b3 = 50. For node 1, jump directly to node 3 with cost a1 × b3 = 10.In the second example, node 3 and node 4 are leaves, so the cost is 0. For node 2, jump to node 4 with cost a2 × b4 = 100. For node 1, jump to node 2 with cost a1 × b2 =  - 400 followed by a jump from 2 to 4 with cost a2 × b4 = 100.


Given a string s, find the number of ways to split s to substrings such that if there are k substrings (p1, p2, p3, ..., pk) in partition, then pi = pk - i + 1 for all i (1 ≤ i ≤ k) and k is even.Since the number of ways can be large, print it modulo 109 + 7.

The only line of input contains a string s (2 ≤ |s| ≤ 106) of even length consisting of lowercase Latin letters.

Print one integer, the number of ways of partitioning the string modulo 109 + 7.

NoteIn the first case, the only way to partition the string is ab|cd|cd|ab.In the second case, the string can be partitioned as ab|b|ab|ab|ab|ab|b|ab or ab|b|abab|abab|b|ab or abbab|ab|ab|abbab.


A dragon symbolizes wisdom, power and wealth. On Lunar New Year's Day, people model a dragon with bamboo strips and clothes, raise them with rods, and hold the rods high and low to resemble a flying dragon.A performer holding the rod low is represented by a 1, while one holding it high is represented by a 2. Thus, the line of performers can be represented by a sequence a1, a2, ..., an.Little Tommy is among them. He would like to choose an interval [l, r] (1 ≤ l ≤ r ≤ n), then reverse al, al + 1, ..., ar so that the length of the longest non-decreasing subsequence of the new sequence is maximum.A non-decreasing subsequence is a sequence of indices p1, p2, ..., pk, such that p1 < p2 < ... < pk and ap1 ≤ ap2 ≤ ... ≤ apk. The length of the subsequence is k.

The first line contains an integer n (1 ≤ n ≤ 2000), denoting the length of the original sequence. The second line contains n space-separated integers, describing the original sequence a1, a2, ..., an (1 ≤ ai ≤ 2, i = 1, 2, ..., n).

Print a single integer, which means the maximum possible length of the longest non-decreasing subsequence of the new sequence.

NoteIn the first example, after reversing [2, 3], the array will become [1, 1, 2, 2], where the length of the longest non-decreasing subsequence is 4.In the second example, after reversing [3, 7], the array will become [1, 1, 1, 1, 2, 2, 2, 2, 2, 1], where the length of the longest non-decreasing subsequence is 9.


You have a team of N people. For a particular task, you can pick any non-empty subset of people. The cost of having x people for the task is xk. Output the sum of costs over all non-empty subsets of people.

Only line of input contains two integers N (1 ≤ N ≤ 109) representing total number of people and k (1 ≤ k ≤ 5000).

Output the sum of costs for all non empty subsets modulo 109 + 7.

NoteIn the first example, there is only one non-empty subset {1} with cost 11 = 1.In the second example, there are seven non-empty subsets.- {1} with cost 12 = 1- {2} with cost 12 = 1- {1, 2} with cost 22 = 4- {3} with cost 12 = 1- {1, 3} with cost 22 = 4- {2, 3} with cost 22 = 4- {1, 2, 3} with cost 32 = 9The total cost is 1 + 1 + 4 + 1 + 4 + 4 + 9 = 24.


In order to put away old things and welcome a fresh new year, a thorough cleaning of the house is a must.Little Tommy finds an old polynomial and cleaned it up by taking it modulo another. But now he regrets doing this...Given two integers p and k, find a polynomial f(x) with non-negative integer coefficients strictly less than k, whose remainder is p when divided by (x + k). That is, f(x) = q(x)·(x + k) + p, where q(x) is a polynomial (not necessarily with integer coefficients).

The only line of input contains two space-separated integers p and k (1 ≤ p ≤ 1018, 2 ≤ k ≤ 2 000).

If the polynomial does not exist, print a single integer -1, or output two lines otherwise. In the first line print a non-negative integer d — the number of coefficients in the polynomial. In the second line print d space-separated integers a0, a1, ..., ad - 1, describing a polynomial  fulfilling the given requirements. Your output should satisfy 0 ≤ ai < k for all 0 ≤ i ≤ d - 1, and ad - 1 ≠ 0. If there are many possible solutions, print any of them.

NoteIn the first example, f(x) = x6 + x5 + x4 + x = (x5 - x4 + 3x3 - 6x2 + 12x - 23)·(x + 2) + 46.In the second example, f(x) = x2 + 205x + 92 = (x - 9)·(x + 214) + 2018.


Firecrackers scare Nian the monster, but they're wayyyyy too noisy! Maybe fireworks make a nice complement.Little Tommy is watching a firework show. As circular shapes spread across the sky, a splendid view unfolds on the night of Lunar New Year's eve.A wonder strikes Tommy. How many regions are formed by the circles on the sky? We consider the sky as a flat plane. A region is a connected part of the plane with positive area, whose bound consists of parts of bounds of the circles and is a curve or several curves without self-intersections, and that does not contain any curve other than its boundaries. Note that exactly one of the regions extends infinitely.

The first line of input contains one integer n (1 ≤ n ≤ 3), denoting the number of circles. The following n lines each contains three space-separated integers x, y and r ( - 10 ≤ x, y ≤ 10, 1 ≤ r ≤ 10), describing a circle whose center is (x, y) and the radius is r. No two circles have the same x, y and r at the same time.

Print a single integer — the number of regions on the plane.

NoteFor the first example,  For the second example,  For the third example,


East or west, home is best. That's why family reunion, the indispensable necessity of Lunar New Year celebration, is put in such a position.After the reunion dinner, Little Tommy plays a game with the family. Here is a concise introduction to this game:   There is a sequence of n non-negative integers p1, p2, ..., pn in the beginning. It is ruled that each integer in this sequence should be non-negative at any time.  You can select two consecutive positive integers in this sequence, pi and pi + 1 (1 ≤ i < n), and then decrease them by their minimum (i. e. min(pi, pi + 1)), the cost of this operation is equal to min(pi, pi + 1). We call such operation as a descension.  The game immediately ends when there are no two consecutive positive integers. Your task is to end the game so that the total cost of your operations is as small as possible. Obviously, every game ends after at most n - 1 descensions. Please share your solution of this game with the lowest cost.

The first line contains one integer n (1 ≤ n ≤ 3·105). The second line contains n space-separated integers p1, p2, ..., pn (0 ≤ pi ≤ 109, i = 1, 2, ..., n).

In the first line print one integer as the number of descensions m (0 ≤ m ≤ n - 1). In the next m lines print the descensions chronologically. More precisely, in each line of the next m lines print one integer i (1 ≤ i < n) representing a descension would operate on pi and pi + 1 such that all the descensions could be utilized from top to bottom. If there are many possible solutions to reach the minimal cost, print any of them.

NoteIn the first sample, one possible best solution is , of which the cost is 1 + 1 = 2.In the second sample, one possible best solution is , of which the cost is 1 + 1 + 1 = 3.


Everything red frightens Nian the monster. So do red paper and... you, red on Codeforces, potential or real.Big Banban has got a piece of paper with endless lattice points, where lattice points form squares with the same area. His most favorite closed shape is the circle because of its beauty and simplicity. Once he had obtained this piece of paper, he prepares it for paper-cutting.  He drew n concentric circles on it and numbered these circles from 1 to n such that the center of each circle is the same lattice point and the radius of the k-th circle is  times the length of a lattice edge.Define the degree of beauty of a lattice point as the summation of the indices of circles such that this lattice point is inside them, or on their bounds. Banban wanted to ask you the total degree of beauty of all the lattice points, but changed his mind.Defining the total degree of beauty of all the lattice points on a piece of paper with n circles as f(n), you are asked to figure out .

The first line contains one integer m (1 ≤ m ≤ 1012).

In the first line print one integer representing .

NoteA piece of paper with 5 circles is shown in the following.  There are 5 types of lattice points where the degree of beauty of each red point is 1 + 2 + 3 + 4 + 5 = 15, the degree of beauty of each orange point is 2 + 3 + 4 + 5 = 14, the degree of beauty of each green point is 4 + 5 = 9, the degree of beauty of each blue point is 5 and the degree of beauty of each gray point is 0. Therefore, f(5) = 5·15 + 4·14 + 4·9 + 8·5 = 207.Similarly, f(1) = 5, f(2) = 23, f(3) = 50, f(4) = 102 and consequently .


Nian is a monster which lives deep in the oceans. Once a year, it shows up on the land, devouring livestock and even people. In order to keep the monster away, people fill their villages with red colour, light, and cracking noise, all of which frighten the monster out of coming.Little Tommy has n lanterns and Big Banban has m lanterns. Tommy's lanterns have brightness a1, a2, ..., an, and Banban's have brightness b1, b2, ..., bm respectively.Tommy intends to hide one of his lanterns, then Banban picks one of Tommy's non-hidden lanterns and one of his own lanterns to form a pair. The pair's brightness will be the product of the brightness of two lanterns.Tommy wants to make the product as small as possible, while Banban tries to make it as large as possible.You are asked to find the brightness of the chosen pair if both of them choose optimally.

The first line contains two space-separated integers n and m (2 ≤ n, m ≤ 50). The second line contains n space-separated integers a1, a2, ..., an. The third line contains m space-separated integers b1, b2, ..., bm. All the integers range from  - 109 to 109.

Print a single integer — the brightness of the chosen pair.

NoteIn the first example, Tommy will hide 20 and Banban will choose 18 from Tommy and 14 from himself.In the second example, Tommy will hide 3 and Banban will choose 2 from Tommy and 1 from himself.


Fafa owns a company that works on huge projects. There are n employees in Fafa's company. Whenever the company has a new project to start working on, Fafa has to divide the tasks of this project among all the employees.Fafa finds doing this every time is very tiring for him. So, he decided to choose the best l employees in his company as team leaders. Whenever there is a new project, Fafa will divide the tasks among only the team leaders and each team leader will be responsible of some positive number of employees to give them the tasks. To make this process fair for the team leaders, each one of them should be responsible for the same number of employees. Moreover, every employee, who is not a team leader, has to be under the responsibility of exactly one team leader, and no team leader is responsible for another team leader.Given the number of employees n, find in how many ways Fafa could choose the number of team leaders l in such a way that it is possible to divide employees between them evenly.

The input consists of a single line containing a positive integer n (2 ≤ n ≤ 105) — the number of employees in Fafa's company.

Print a single integer representing the answer to the problem.

NoteIn the second sample Fafa has 3 ways:  choose only 1 employee as a team leader with 9 employees under his responsibility.  choose 2 employees as team leaders with 4 employees under the responsibility of each of them.  choose 5 employees as team leaders with 1 employee under the responsibility of each of them.


Fifa and Fafa are sharing a flat. Fifa loves video games and wants to download a new soccer game. Unfortunately, Fafa heavily uses the internet which consumes the quota. Fifa can access the internet through his Wi-Fi access point. This access point can be accessed within a range of r meters (this range can be chosen by Fifa) from its position. Fifa must put the access point inside the flat which has a circular shape of radius R. Fifa wants to minimize the area that is not covered by the access point inside the flat without letting Fafa or anyone outside the flat to get access to the internet.The world is represented as an infinite 2D plane. The flat is centered at (x1, y1) and has radius R and Fafa's laptop is located at (x2, y2), not necessarily inside the flat. Find the position and the radius chosen by Fifa for his access point which minimizes the uncovered area.

The single line of the input contains 5 space-separated integers R, x1, y1, x2, y2 (1 ≤ R ≤ 105, |x1|, |y1|, |x2|, |y2| ≤ 105).

Print three space-separated numbers xap, yap, r where (xap, yap) is the position which Fifa chose for the access point and r is the radius of its range.  Your answer will be considered correct if the radius does not differ from optimal more than 10 - 6 absolutely or relatively, and also the radius you printed can be changed by no more than 10 - 6 (absolutely or relatively) in such a way that all points outside the flat and Fafa's laptop position are outside circle of the access point range.


Apart from Nian, there is a daemon named Sui, which terrifies children and causes them to become sick. Parents give their children money wrapped in red packets and put them under the pillow, so that when Sui tries to approach them, it will be driven away by the fairies inside.Big Banban is hesitating over the amount of money to give out. He considers loops to be lucky since it symbolizes unity and harmony.He would like to find a positive integer n not greater than 1018, such that there are exactly k loops in the decimal representation of n, or determine that such n does not exist.A loop is a planar area enclosed by lines in the digits' decimal representation written in Arabic numerals. For example, there is one loop in digit 4, two loops in 8 and no loops in 5. Refer to the figure below for all exact forms.

The first and only line contains an integer k (1 ≤ k ≤ 106) — the desired number of loops.

Output an integer — if no such n exists, output -1; otherwise output any such n. In the latter case, your output should be a positive decimal integer not exceeding 1018.


Ancient Egyptians are known to have understood difficult concepts in mathematics. The ancient Egyptian mathematician Ahmes liked to write a kind of arithmetic expressions on papyrus paper which he called as Ahmes arithmetic expression.An Ahmes arithmetic expression can be defined as:   "d" is an Ahmes arithmetic expression, where d is a one-digit positive integer;  "(E1 op E2)" is an Ahmes arithmetic expression, where E1 and E2 are valid Ahmes arithmetic expressions (without spaces) and op is either plus ( + ) or minus ( - ).  For example 5, (1-1) and ((1+(2-3))-5) are valid Ahmes arithmetic expressions.On his trip to Egypt, Fafa found a piece of papyrus paper having one of these Ahmes arithmetic expressions written on it. Being very ancient, the papyrus piece was very worn out. As a result, all the operators were erased, keeping only the numbers and the brackets. Since Fafa loves mathematics, he decided to challenge himself with the following task:Given the number of plus and minus operators in the original expression, find out the maximum possible value for the expression on the papyrus paper after putting the plus and minus operators in the place of the original erased operators.

The first line contains a string E (1 ≤ |E| ≤ 104) — a valid Ahmes arithmetic expression. All operators are erased and replaced with '?'. The second line contains two space-separated integers P and M (0 ≤ min(P, M) ≤ 100) — the number of plus and minus operators, respectively.  It is guaranteed that P + M =  the number of erased operators.

Print one line containing the answer to the problem.

Note  The first sample will be (1 + 1)  =  2.  The second sample will be (2 + (1 - 2))  =  1.  The third sample will be ((1 - (5 - 7)) + ((6 + 2) + 7))  =  18.  The fourth sample will be ((1 + (5 + 7)) - ((6 - 2) - 7))  =  16.


Ancient Egyptians are known to have used a large set of symbols  to write on the walls of the temples. Fafa and Fifa went to one of the temples and found two non-empty words S1 and S2 of equal lengths on the wall of temple written one below the other. Since this temple is very ancient, some symbols from the words were erased. The symbols in the set  have equal probability for being in the position of any erased symbol.Fifa challenged Fafa to calculate the probability that S1 is lexicographically greater than S2. Can you help Fafa with this task?You know that , i. e. there were m distinct characters in Egyptians' alphabet, in this problem these characters are denoted by integers from 1 to m in alphabet order. A word x is lexicographically greater than a word y of the same length, if the words are same up to some position, and then the word x has a larger character, than the word y.We can prove that the probability equals to some fraction , where P and Q are coprime integers, and . Print as the answer the value , i. e. such a non-negative integer less than 109 + 7, such that , where  means that a and b give the same remainders when divided by m.

The first line contains two integers n and m (1 ≤ n,  m ≤ 105) — the length of each of the two words and the size of the alphabet , respectively. The second line contains n integers a1, a2, ..., an (0 ≤ ai ≤ m) — the symbols of S1. If ai = 0, then the symbol at position i was erased. The third line contains n integers representing S2 with the same format as S1.

Print the value , where P and Q are coprime and  is the answer to the problem.

NoteIn the first sample, the first word can be converted into (1) or (2). The second option is the only one that will make it lexicographically larger than the second word. So, the answer to the problem will be , that is 500000004, because .In the second example, there is no replacement for the zero in the second word that will make the first one lexicographically larger. So, the answer to the problem is , that is 0.


Fafa has an array A of n positive integers, the function f(A) is defined as . He wants to do q queries of two types:  1 l r x — find the maximum possible value of f(A), if x is to be added to one element in the range [l,  r]. You can choose to which element to add x.  2 l r x — increase all the elements in the range [l,  r] by value x. Note that queries of type 1 don't affect the array elements.

The first line contains one integer n (3 ≤ n ≤ 105) — the length of the array. The second line contains n positive integers a1, a2, ..., an (0 < ai ≤ 109) — the array elements. The third line contains an integer q (1 ≤ q ≤ 105) — the number of queries.  Then q lines follow, line i describes the i-th query and contains four integers ti li ri xi .  It is guaranteed that at least one of the queries is of type 1.

For each query of type 1, print the answer to the query.


Julia is going to cook a chicken in the kitchen of her dormitory. To save energy, the stove in the kitchen automatically turns off after k minutes after turning on.During cooking, Julia goes to the kitchen every d minutes and turns on the stove if it is turned off. While the cooker is turned off, it stays warm. The stove switches on and off instantly.It is known that the chicken needs t minutes to be cooked on the stove, if it is turned on, and 2t minutes, if it is turned off. You need to find out, how much time will Julia have to cook the chicken, if it is considered that the chicken is cooked evenly, with constant speed when the stove is turned on and at a constant speed when it is turned off.

The single line contains three integers k, d and t (1 ≤ k, d, t ≤ 1018).

Print a single number, the total time of cooking in minutes. The relative or absolute error must not exceed 10 - 9. Namely, let's assume that your answer is x and the answer of the jury is y. The checker program will consider your answer correct if .

NoteIn the first example, the chicken will be cooked for 3 minutes on the turned on stove, after this it will be cooked for . Then the chicken will be cooked for one minute on a turned off stove, it will be cooked for . Thus, after four minutes the chicken will be cooked for . Before the fifth minute Julia will turn on the stove and after 2.5 minutes the chicken will be ready .In the second example, when the stove is turned off, Julia will immediately turn it on, so the stove will always be turned on and the chicken will be cooked in 20 minutes.


Welcome to another task about breaking the code lock! Explorers Whitfield and Martin came across an unusual safe, inside of which, according to rumors, there are untold riches, among which one can find the solution of the problem of discrete logarithm!Of course, there is a code lock is installed on the safe. The lock has a screen that displays a string of n lowercase Latin letters. Initially, the screen displays string s. Whitfield and Martin found out that the safe will open when string t will be displayed on the screen.The string on the screen can be changed using the operation «shift x». In order to apply this operation, explorers choose an integer x from 0 to n inclusive. After that, the current string p = αβ changes to βRα, where the length of β is x, and the length of α is n - x. In other words, the suffix of the length x of string p is reversed and moved to the beginning of the string. For example, after the operation «shift 4» the string «abcacb» will be changed with string «bcacab », since α = ab, β = cacb, βR = bcac.Explorers are afraid that if they apply too many operations «shift», the lock will be locked forever. They ask you to find a way to get the string t on the screen, using no more than 6100 operations.

The first line contains an integer n, the length of the strings s and t (1 ≤ n ≤ 2 000). After that, there are two strings s and t, consisting of n lowercase Latin letters each.

If it is impossible to get string t from string s using no more than 6100 operations «shift», print a single number  - 1. Otherwise, in the first line output the number of operations k (0 ≤ k ≤ 6100). In the next line output k numbers xi corresponding to the operations «shift xi» (0 ≤ xi ≤ n) in the order in which they should be applied.


Petya and Vasya arranged a game. The game runs by the following rules. Players have a directed graph consisting of n vertices and m edges. One of the vertices contains a chip. Initially the chip is located at vertex s. Players take turns moving the chip along some edge of the graph. Petya goes first. Player who can't move the chip loses. If the game lasts for 106 turns the draw is announced.Vasya was performing big laboratory work in "Spelling and parts of speech" at night before the game, so he fell asleep at the very beginning of the game. Petya decided to take the advantage of this situation and make both Petya's and Vasya's moves.Your task is to help Petya find out if he can win the game or at least draw a tie.

The first line of input contain two integers n and m — the number of vertices and the number of edges in the graph (2 ≤ n ≤ 105, 0 ≤ m ≤ 2·105). The next n lines contain the information about edges of the graph. i-th line (1 ≤ i ≤ n) contains nonnegative integer ci — number of vertices such that there is an edge from i to these vertices and ci distinct integers ai, j — indices of these vertices (1 ≤ ai, j ≤ n, ai, j ≠ i). It is guaranteed that the total sum of ci equals to m. The next line contains index of vertex s — the initial position of the chip (1 ≤ s ≤ n).

If Petya can win print «Win» in the first line. In the next line print numbers v1, v2, ..., vk (1 ≤ k ≤ 106) — the sequence of vertices Petya should visit for the winning. Vertex v1 should coincide with s. For i = 1... k - 1 there should be an edge from vi to vi + 1 in the graph. There must be no possible move from vertex vk. The sequence should be such that Petya wins the game. If Petya can't win but can draw a tie, print «Draw» in the only line. Otherwise print «Lose».

NoteIn the first example the graph is the following:  Initially the chip is located at vertex 1. In the first move Petya moves the chip to vertex 2, after that he moves it to vertex 4 for Vasya. After that he moves to vertex 5. Now it is Vasya's turn and there is no possible move, so Petya wins.In the second example the graph is the following:  Initially the chip is located at vertex 2. The only possible Petya's move is to go to vertex 1. After that he has to go to 3 for Vasya. Now it's Petya's turn but he has no possible move, so Petya loses.In the third example the graph is the following:  Petya can't win, but he can move along the cycle, so the players will draw a tie.


Vitya loves programming and problem solving, but sometimes, to distract himself a little, he plays computer games. Once he found a new interesting game about tanks, and he liked it so much that he went through almost all levels in one day. Remained only the last level, which was too tricky. Then Vitya remembered that he is a programmer, and wrote a program that helped him to pass this difficult level. Try do the same.The game is organized as follows. There is a long road, two cells wide and n cells long. Some cells have obstacles. You control a tank that occupies one cell. Initially, the tank is located before the start of the road, in a cell with coordinates (0, 1). Your task is to move the tank to the end of the road, to the cell (n + 1, 1) or (n + 1, 2).  Every second the tank moves one cell to the right: the coordinate x is increased by one. When you press the up or down arrow keys, the tank instantly changes the lane, that is, the y coordinate. When you press the spacebar, the tank shoots, and the nearest obstacle along the lane in which the tank rides is instantly destroyed. In order to load a gun, the tank needs t seconds. Initially, the gun is not loaded, that means, the first shot can be made only after t seconds after the tank starts to move.If at some point the tank is in the same cell with an obstacle not yet destroyed, it burns out. If you press the arrow exactly at the moment when the tank moves forward, the tank will first move forward, and then change the lane, so it will not be possible to move diagonally.Your task is to find out whether it is possible to pass the level, and if possible, to find the order of actions the player need to make.

The first line contains four integers n, m1, m2 and t, the length of the field, the number of obstacles in the first lane, the number of obstacles in the second lane and the number of tank steps before reloading, respectively (1 ≤ n ≤ 109; 0 ≤ m1, m2 ≤ n; 0 ≤ m1 + m2 ≤ 106; 1 ≤ t ≤ n). The next two lines contain a description of the obstacles. The first of these lines contains m1 numbers xi — the obstacle coordinates in the first lane (1 ≤ xi ≤ n; xi < xi + 1). The y coordinate for all these obstacles will be 1. The second line contains m2 numbers describing the obstacles of the second lane in the same format. The y coordinate of all these obstacles will be 2.

In the first line print «Yes», if it is possible to pass the level, or «No», otherwise. If it is possible, then in the second line print the number of times the tank moves from one lane to another, and in the next line print the coordinates of the transitions, one number per transition: the coordinate x (0 ≤ x ≤ n + 1). All transition coordinates coordinates must be distinct and should be output in strictly increasing order.The number of transitions should not exceed 2·106. If the tank can pass the level, then it can do it using no more than 2·106 transitions. In the fourth line print the number of shots that the tank makes during the movement, in the following lines print two numbers, x and y coordinates of the point (1 ≤ x ≤ n, 1 ≤ y ≤ 2), from which the tank fired a shot, the number of shots must not exceed m1 + m2. Shots must be output in the order in which they are fired. If there are several solutions, output any one.


The recent All-Berland Olympiad in Informatics featured n participants with each scoring a certain amount of points.As the head of the programming committee, you are to determine the set of participants to be awarded with diplomas with respect to the following criteria:   At least one participant should get a diploma.  None of those with score equal to zero should get awarded.  When someone is awarded, all participants with score not less than his score should also be awarded. Determine the number of ways to choose a subset of participants that will receive the diplomas.

The first line contains a single integer n (1 ≤ n ≤ 100) — the number of participants. The next line contains a sequence of n integers a1, a2, ..., an (0 ≤ ai ≤ 600) — participants' scores. It's guaranteed that at least one participant has non-zero score.

Print a single integer — the desired number of ways.

NoteThere are three ways to choose a subset in sample case one.  Only participants with 3 points will get diplomas.  Participants with 2 or 3 points will get diplomas.  Everyone will get a diploma! The only option in sample case two is to award everyone.Note that in sample case three participants with zero scores cannot get anything.


The weather is fine today and hence it's high time to climb the nearby pine and enjoy the landscape.The pine's trunk includes several branches, located one above another and numbered from 2 to y. Some of them (more precise, from 2 to p) are occupied by tiny vile grasshoppers which you're at war with. These grasshoppers are known for their awesome jumping skills: the grasshopper at branch x can jump to branches .Keeping this in mind, you wisely decided to choose such a branch that none of the grasshoppers could interrupt you. At the same time you wanna settle as high as possible since the view from up there is simply breathtaking.In other words, your goal is to find the highest branch that cannot be reached by any of the grasshoppers or report that it's impossible.

The only line contains two integers p and y (2 ≤ p ≤ y ≤ 109).

Output the number of the highest suitable branch. If there are none, print -1 instead.

NoteIn the first sample case grasshopper from branch 2 reaches branches 2, 4 and 6 while branch 3 is initially settled by another grasshopper. Therefore the answer is 5.It immediately follows that there are no valid branches in second sample case.


Victor tries to write his own text editor, with word correction included. However, the rules of word correction are really strange.Victor thinks that if a word contains two consecutive vowels, then it's kinda weird and it needs to be replaced. So the word corrector works in such a way: as long as there are two consecutive vowels in the word, it deletes the first vowel in a word such that there is another vowel right before it. If there are no two consecutive vowels in the word, it is considered to be correct.You are given a word s. Can you predict what will it become after correction?In this problem letters a, e, i, o, u and y are considered to be vowels.

The first line contains one integer n (1 ≤ n ≤ 100) — the number of letters in word s before the correction. The second line contains a string s consisting of exactly n lowercase Latin letters — the word before the correction.

NoteExplanations of the examples:  There is only one replace: weird  werd; No replace needed since there are no two consecutive vowels; aaeaa  aeaa  aaa  aa  a.


Two neighboring kingdoms decided to build a wall between them with some gates to enable the citizens to go from one kingdom to another. Each time a citizen passes through a gate, he has to pay one silver coin.The world can be represented by the first quadrant of a plane and the wall is built along the identity line (i.e. the line with the equation x = y). Any point below the wall belongs to the first kingdom while any point above the wall belongs to the second kingdom. There is a gate at any integer point on the line (i.e. at points (0, 0), (1, 1), (2, 2), ...). The wall and the gates do not belong to any of the kingdoms. Fafa is at the gate at position (0, 0) and he wants to walk around in the two kingdoms. He knows the sequence S of moves he will do. This sequence is a string where each character represents a move. The two possible moves Fafa will do are 'U' (move one step up, from (x, y) to (x, y + 1)) and 'R' (move one step right, from (x, y) to (x + 1, y)). Fafa wants to know the number of silver coins he needs to pay to walk around the two kingdoms following the sequence S. Note that if Fafa visits a gate without moving from one kingdom to another, he pays no silver coins. Also assume that he doesn't pay at the gate at point (0, 0), i. e. he is initially on the side he needs.

The first line of the input contains single integer n (1 ≤ n ≤ 105) — the number of moves in the walking sequence. The second line contains a string S of length n consisting of the characters 'U' and 'R' describing the required moves. Fafa will follow the sequence S in order from left to right.

On a single line, print one integer representing the number of silver coins Fafa needs to pay at the gates to follow the sequence S.

NoteThe figure below describes the third sample. The red arrows represent the sequence of moves Fafa will follow. The green gates represent the gates at which Fafa have to pay silver coins.


You and your friend are participating in a TV show "Run For Your Prize".At the start of the show n prizes are located on a straight line. i-th prize is located at position ai. Positions of all prizes are distinct. You start at position 1, your friend — at position 106 (and there is no prize in any of these two positions). You have to work as a team and collect all prizes in minimum possible time, in any order.You know that it takes exactly 1 second to move from position x to position x + 1 or x - 1, both for you and your friend. You also have trained enough to instantly pick up any prize, if its position is equal to your current position (and the same is true for your friend). Carrying prizes does not affect your speed (or your friend's speed) at all.Now you may discuss your strategy with your friend and decide who will pick up each prize. Remember that every prize must be picked up, either by you or by your friend.What is the minimum number of seconds it will take to pick up all the prizes?

The first line contains one integer n (1 ≤ n ≤ 105) — the number of prizes. The second line contains n integers a1, a2, ..., an (2 ≤ ai ≤ 106 - 1) — the positions of the prizes. No two prizes are located at the same position. Positions are given in ascending order.

Print one integer — the minimum number of seconds it will take to collect all prizes.

NoteIn the first example you take all the prizes: take the first at 1, the second at 2 and the third at 8.In the second example you take the first prize in 1 second and your friend takes the other in 5 seconds, you do this simultaneously, so the total time is 5.


Let's denote a m-free matrix as a binary (that is, consisting of only 1's and 0's) matrix such that every square submatrix of size m × m of this matrix contains at least one zero. Consider the following problem:You are given two integers n and m. You have to construct an m-free square matrix of size n × n such that the number of 1's in this matrix is maximum possible. Print the maximum possible number of 1's in such matrix.You don't have to solve this problem. Instead, you have to construct a few tests for it.You will be given t numbers x1, x2, ..., xt. For every , find two integers ni and mi (ni ≥ mi) such that the answer for the aforementioned problem is exactly xi if we set n = ni and m = mi.

The first line contains one integer t (1 ≤ t ≤ 100) — the number of tests you have to construct. Then t lines follow, i-th line containing one integer xi (0 ≤ xi ≤ 109). Note that in hacks you have to set t = 1.

For each test you have to construct, output two positive numbers ni and mi (1 ≤ mi ≤ ni ≤ 109) such that the maximum number of 1's in a mi-free ni × ni matrix is exactly xi. If there are multiple solutions, you may output any of them; and if this is impossible to construct a test, output a single integer  - 1.


Musicians of a popular band "Flayer" have announced that they are going to "make their exit" with a world tour. Of course, they will visit Berland as well.There are n cities in Berland. People can travel between cities using two-directional train routes; there are exactly m routes, i-th route can be used to go from city vi to city ui (and from ui to vi), and it costs wi coins to use this route.Each city will be visited by "Flayer", and the cost of the concert ticket in i-th city is ai coins.You have friends in every city of Berland, and they, knowing about your programming skills, asked you to calculate the minimum possible number of coins they have to pay to visit the concert. For every city i you have to compute the minimum number of coins a person from city i has to spend to travel to some city j (or possibly stay in city i), attend a concert there, and return to city i (if j ≠ i).Formally, for every  you have to calculate , where d(i, j) is the minimum number of coins you have to spend to travel from city i to city j. If there is no way to reach city j from city i, then we consider d(i, j) to be infinitely large.

The first line contains two integers n and m (2 ≤ n ≤ 2·105, 1 ≤ m ≤ 2·105). Then m lines follow, i-th contains three integers vi, ui and wi (1 ≤ vi, ui ≤ n, vi ≠ ui, 1 ≤ wi ≤ 1012) denoting i-th train route. There are no multiple train routes connecting the same pair of cities, that is, for each (v, u) neither extra (v, u) nor (u, v) present in input. The next line contains n integers a1, a2, ... ak (1 ≤ ai ≤ 1012) — price to attend the concert in i-th city.

Print n integers. i-th of them must be equal to the minimum number of coins a person from city i has to spend to travel to some city j (or possibly stay in city i), attend a concert there, and return to city i (if j ≠ i).


You are given an array a of length n. We define fa the following way:  Initially fa = 0, M = 1;  for every 2 ≤ i ≤ n if aM < ai then we set fa = fa + aM and then set M = i. Calculate the sum of fa over all n! permutations of the array a modulo 109 + 7.Note: two elements are considered different if their indices differ, so for every array a there are exactly n! permutations.

The first line contains integer n (1 ≤ n ≤  1 000 000) — the size of array a. Second line contains n integers a1, a2, ..., an (1 ≤  ai ≤  109).

Print the only integer, the sum of fa over all n! permutations of the array a modulo 109 + 7.

NoteFor the second example all the permutations are:  p = [1, 2, 3] : fa is equal to 1;  p = [1, 3, 2] : fa is equal to 1;  p = [2, 1, 3] : fa is equal to 1;  p = [2, 3, 1] : fa is equal to 1;  p = [3, 1, 2] : fa is equal to 0;  p = [3, 2, 1] : fa is equal to 0. Where p is the array of the indices of initial array a. The sum of fa is equal to 4.


You are given an undirected connected graph with weighted edges. The length of some path between two vertices is the bitwise xor of weights of all edges belonging to this path (if some edge is traversed more than once, then it is included in bitwise xor the same number of times). There are three types of queries you have to process:  1 x y d — add an edge connecting vertex x to vertex y with weight d. It is guaranteed that there is no edge connecting x to y before this query;  2 x y — remove an edge connecting vertex x to vertex y. It is guaranteed that there was such edge in the graph, and the graph stays connected after this query;  3 x y — calculate the length of the shortest path (possibly non-simple) from vertex x to vertex y. Print the answers for all queries of type 3.

The first line contains two numbers n and m (1 ≤ n, m ≤ 200000) — the number of vertices and the number of edges in the graph, respectively. Then m lines follow denoting the edges of the graph. Each line contains three integers x, y and d (1 ≤ x < y ≤ n, 0 ≤ d ≤ 230 - 1). Each pair (x, y) is listed at most once. The initial graph is connected. Then one line follows, containing an integer q (1 ≤ q ≤ 200000) — the number of queries you have to process. Then q lines follow, denoting queries in the following form:   1 x y d (1 ≤ x < y ≤ n, 0 ≤ d ≤ 230 - 1) — add an edge connecting vertex x to vertex y with weight d. It is guaranteed that there is no edge connecting x to y before this query;  2 x y (1 ≤ x < y ≤ n) — remove an edge connecting vertex x to vertex y. It is guaranteed that there was such edge in the graph, and the graph stays connected after this query;  3 x y (1 ≤ x < y ≤ n) — calculate the length of the shortest path (possibly non-simple) from vertex x to vertex y.  It is guaranteed that at least one query has type 3.

Print the answers for all queries of type 3 in the order they appear in input.


As you could know there are no male planes nor female planes. However, each plane on Earth likes some other plane. There are n planes on Earth, numbered from 1 to n, and the plane with number i likes the plane with number fi, where 1 ≤ fi ≤ n and fi ≠ i.We call a love triangle a situation in which plane A likes plane B, plane B likes plane C and plane C likes plane A. Find out if there is any love triangle on Earth.

The first line contains a single integer n (2 ≤ n ≤ 5000) — the number of planes. The second line contains n integers f1, f2, ..., fn (1 ≤ fi ≤ n, fi ≠ i), meaning that the i-th plane likes the fi-th.

Output «YES» if there is a love triangle consisting of planes on Earth. Otherwise, output «NO». You can output any letter in lower case or in upper case.

NoteIn first example plane 2 likes plane 4, plane 4 likes plane 1, plane 1 likes plane 2 and that is a love triangle.In second example there are no love triangles.


Dima has a hamsters farm. Soon N hamsters will grow up on it and Dima will sell them in a city nearby.Hamsters should be transported in boxes. If some box is not completely full, the hamsters in it are bored, that's why each box should be completely full with hamsters.Dima can buy boxes at a factory. The factory produces boxes of K kinds, boxes of the i-th kind can contain in themselves ai hamsters. Dima can buy any amount of boxes, but he should buy boxes of only one kind to get a wholesale discount.Of course, Dima would buy boxes in such a way that each box can be completely filled with hamsters and transported to the city. If there is no place for some hamsters, Dima will leave them on the farm.Find out how many boxes and of which type should Dima buy to transport maximum number of hamsters.

The first line contains two integers N and K (0 ≤ N ≤ 1018, 1 ≤ K ≤ 105) — the number of hamsters that will grow up on Dima's farm and the number of types of boxes that the factory produces. The second line contains K integers a1, a2, ..., aK (1 ≤ ai ≤ 1018 for all i) — the capacities of boxes.

Output two integers: the type of boxes that Dima should buy and the number of boxes of that type Dima should buy. Types of boxes are numbered from 1 to K in the order they are given in input. If there are many correct answers, output any of them.


In distant future on Earth day lasts for n hours and that's why there are n timezones. Local times in adjacent timezones differ by one hour. For describing local time, hours numbers from 1 to n are used, i.e. there is no time "0 hours", instead of it "n hours" is used. When local time in the 1-st timezone is 1 hour, local time in the i-th timezone is i hours.Some online programming contests platform wants to conduct a contest that lasts for an hour in such a way that its beginning coincides with beginning of some hour (in all time zones). The platform knows, that there are ai people from i-th timezone who want to participate in the contest. Each person will participate if and only if the contest starts no earlier than s hours 00 minutes local time and ends not later than f hours 00 minutes local time. Values s and f are equal for all time zones. If the contest starts at f hours 00 minutes local time, the person won't participate in it.Help platform select such an hour, that the number of people who will participate in the contest is maximum.

The first line contains a single integer n (2 ≤ n ≤ 100 000) — the number of hours in day. The second line contains n space-separated integers a1, a2, ..., an (1 ≤ ai ≤ 10 000), where ai is the number of people in the i-th timezone who want to participate in the contest. The third line contains two space-separated integers s and f (1 ≤ s < f ≤ n).

Output a single integer — the time of the beginning of the contest (in the first timezone local time), such that the number of participants will be maximum possible. If there are many answers, output the smallest among them.

NoteIn the first example, it's optimal to start competition at 3 hours (in first timezone). In this case, it will be 1 hour in the second timezone and 2 hours in the third timezone. Only one person from the first timezone won't participate.In second example only people from the third and the fourth timezones will participate.


You are given a string s, initially consisting of n lowercase Latin letters. After that, you perform k operations with it, where . During i-th operation you must erase some substring of length exactly 2i - 1 from s.Print the lexicographically minimal string you may obtain after performing k such operations.

The only line contains one string s consisting of n lowercase Latin letters (1 ≤ n ≤ 5000).

Print the lexicographically minimal string you may obtain after performing k operations.

NotePossible operations in examples:  adcbca  adcba  aba;  abacabadabacaba  abcabadabacaba  aabadabacaba  aabacaba.


Valya and Tolya are an ideal pair, but they quarrel sometimes. Recently, Valya took offense at her boyfriend because he came to her in t-shirt with lettering that differs from lettering on her pullover. Now she doesn't want to see him and Tolya is seating at his room and crying at her photos all day long.This story could be very sad but fairy godmother (Tolya's grandmother) decided to help them and restore their relationship. She secretly took Tolya's t-shirt and Valya's pullover and wants to make the letterings on them same. In order to do this, for one unit of mana she can buy a spell that can change some letters on the clothes. Your task is calculate the minimum amount of mana that Tolya's grandmother should spend to rescue love of Tolya and Valya.More formally, letterings on Tolya's t-shirt and Valya's pullover are two strings with same length n consisting only of lowercase English letters. Using one unit of mana, grandmother can buy a spell of form (c1, c2) (where c1 and c2 are some lowercase English letters), which can arbitrary number of times transform a single letter c1 to c2 and vise-versa on both Tolya's t-shirt and Valya's pullover. You should find the minimum amount of mana that grandmother should spend to buy a set of spells that can make the letterings equal. In addition you should output the required set of spells.

The first line contains a single integer n (1 ≤ n ≤ 105) — the length of the letterings. The second line contains a string with length n, consisting of lowercase English letters — the lettering on Valya's pullover. The third line contains the lettering on Tolya's t-shirt in the same format.

In the first line output a single integer — the minimum amount of mana t required for rescuing love of Valya and Tolya. In the next t lines output pairs of space-separated lowercase English letters — spells that Tolya's grandmother should buy. Spells and letters in spells can be printed in any order. If there are many optimal answers, output any.

NoteIn first example it's enough to buy two spells: ('a','d') and ('b','a'). Then first letters will coincide when we will replace letter 'a' with 'd'. Second letters will coincide when we will replace 'b' with 'a'. Third letters will coincide when we will at first replace 'b' with 'a' and then 'a' with 'd'.


You are given a multiset S consisting of positive integers (initially empty). There are two kind of queries:   Add a positive integer to S, the newly added integer is not less than any number in it.  Find a subset s of the set S such that the value  is maximum possible. Here max(s) means maximum value of elements in s,  — the average value of numbers in s. Output this maximum possible value of .

The first line contains a single integer Q (1 ≤ Q ≤ 5·105) — the number of queries. Each of the next Q lines contains a description of query. For queries of type 1 two integers 1 and x are given, where x (1 ≤ x ≤ 109) is a number that you should add to S. It's guaranteed that x is not less than any number in S. For queries of type 2, a single integer 2 is given. It's guaranteed that the first query has type 1, i. e. S is not empty when a query of type 2 comes.

Output the answer for each query of the second type in the order these queries are given in input. Each number should be printed in separate line. Your answer is considered correct, if each of your answers has absolute or relative error not greater than 10 - 6. Formally, let your answer be a, and the jury's answer be b. Your answer is considered correct if .