문제
평화로운 삶을 살던 하이히는 어느 날 갑자기 바이비의 새로운 요세푸스 게임에 참가하게 되었다! 새로운 요세푸스 게임은 다음과 같이 진행된다.
- 게임이 시작하기 전, 양의 정수 $M$이 정해진다.
- 1ドル$번 참가자부터 $N$번 참가자까지 시계 방향으로 차례로 원을 그리며 앉는다.
- 시작할 때는 화살표를 $N$번 참가자를 가리키도록 놓는다.
- 이후 모든 사람이 탈락할 때까지 다음 과정을 반복한다.
- 1ドル$ 이상 $M$ 이하의 정수 $K$가 정해진다. 이 수는 이전과 같을 수도, 다를 수도 있다.
- 화살표를 시계 방향으로 $K$번 돌린다. 이때, 탈락한 참가자의 자리는 건너뛴다.
- 이후 화살표가 가리키는 참가자가 탈락한다. 이로 인해 화살표가 돌아가지는 않는다.
예로, 다음은 $N = 7,ドル $M = 10$일 때 가능한 새로운 요세푸스 게임의 진행 과정이다.
원래라면 어디에 있어야 탈락하지 않고 끝까지 남을 수 있을지 궁금해하는 것이 일반적이지만, 호기심에 가득 찬 하이히는 사람들이 탈락한 순서가 주어질 때 $K$가 최소 몇 번 달라졌는지 구해보기로 했다!
출력
첫째 줄에 게임이 진행되면서 $K$가 최소 몇 번 달라졌는지 출력한다. $K$를 처음 정하는 것은 $K$가 달라지는 것으로 세지 않는다.
[{"problem_id":"34134","problem_lang":"0","title":"[Y] \uc0c8\ub85c\uc6b4 \uc694\uc138\ud478\uc2a4 \ubb38\uc81c","description":"<p>\ud3c9\ud654\ub85c\uc6b4 \uc0b6\uc744 \uc0b4\ub358 \ud558\uc774\ud788\ub294 \uc5b4\ub290 \ub0a0 \uac11\uc790\uae30 \ubc14\uc774\ube44\uc758 \uc0c8\ub85c\uc6b4 \uc694\uc138\ud478\uc2a4 \uac8c\uc784\uc5d0 \ucc38\uac00\ud558\uac8c \ub418\uc5c8\ub2e4! \uc0c8\ub85c\uc6b4 \uc694\uc138\ud478\uc2a4 \uac8c\uc784\uc740 \ub2e4\uc74c\uacfc \uac19\uc774 \uc9c4\ud589\ub41c\ub2e4.<\/p>\r\n\r\n<ol>\r\n\t<li>\uac8c\uc784\uc774 \uc2dc\uc791\ud558\uae30 \uc804, \uc591\uc758 \uc815\uc218 $M$\uc774 \uc815\ud574\uc9c4\ub2e4.<\/li>\r\n\t<li>$1$\ubc88 \ucc38\uac00\uc790\ubd80\ud130 $N$\ubc88 \ucc38\uac00\uc790\uae4c\uc9c0 \uc2dc\uacc4 \ubc29\ud5a5\uc73c\ub85c \ucc28\ub840\ub85c \uc6d0\uc744 \uadf8\ub9ac\uba70 \uc549\ub294\ub2e4.<\/li>\r\n\t<li>\uc2dc\uc791\ud560 \ub54c\ub294 \ud654\uc0b4\ud45c\ub97c $N$\ubc88 \ucc38\uac00\uc790\ub97c \uac00\ub9ac\ud0a4\ub3c4\ub85d \ub193\ub294\ub2e4.<\/li>\r\n\t<li>\uc774\ud6c4 \ubaa8\ub4e0 \uc0ac\ub78c\uc774 \ud0c8\ub77d\ud560 \ub54c\uae4c\uc9c0 \ub2e4\uc74c \uacfc\uc815\uc744 \ubc18\ubcf5\ud55c\ub2e4.\r\n\t<ul>\r\n\t\t<li>$1$ \uc774\uc0c1 $M$ \uc774\ud558\uc758 \uc815\uc218 $K$\uac00 \uc815\ud574\uc9c4\ub2e4. \uc774 \uc218\ub294 \uc774\uc804\uacfc \uac19\uc744 \uc218\ub3c4, \ub2e4\ub97c \uc218\ub3c4 \uc788\ub2e4.<\/li>\r\n\t\t<li>\ud654\uc0b4\ud45c\ub97c \uc2dc\uacc4 \ubc29\ud5a5\uc73c\ub85c $K$\ubc88 \ub3cc\ub9b0\ub2e4. \uc774\ub54c, \ud0c8\ub77d\ud55c \ucc38\uac00\uc790\uc758 \uc790\ub9ac\ub294 \uac74\ub108\ub6f4\ub2e4.<\/li>\r\n\t\t<li>\uc774\ud6c4 \ud654\uc0b4\ud45c\uac00 \uac00\ub9ac\ud0a4\ub294 \ucc38\uac00\uc790\uac00 \ud0c8\ub77d\ud55c\ub2e4. \uc774\ub85c \uc778\ud574 \ud654\uc0b4\ud45c\uac00 \ub3cc\uc544\uac00\uc9c0\ub294 \uc54a\ub294\ub2e4.<\/li>\r\n\t<\/ul>\r\n\t<\/li>\r\n<\/ol>\r\n\r\n<p>\uc608\ub85c, \ub2e4\uc74c\uc740 $N = 7$, $M = 10$\uc77c \ub54c \uac00\ub2a5\ud55c \uc0c8\ub85c\uc6b4 \uc694\uc138\ud478\uc2a4 \uac8c\uc784\uc758 \uc9c4\ud589 \uacfc\uc815\uc774\ub2e4.<\/p>\r\n\r\n<p style=\"text-align: center;\"><img alt=\"\" src=\"https:\/\/upload.acmicpc.net\/e193afc6-3656-4f69-ae24-6d5ed08830ba\/-\/preview\/\" style=\"width: 797px; max-width: 100%\" \/><\/p>\r\n\r\n<p>\uc6d0\ub798\ub77c\uba74 \uc5b4\ub514\uc5d0 \uc788\uc5b4\uc57c \ud0c8\ub77d\ud558\uc9c0 \uc54a\uace0 \ub05d\uae4c\uc9c0 \ub0a8\uc744 \uc218 \uc788\uc744\uc9c0 \uad81\uae08\ud574\ud558\ub294 \uac83\uc774 \uc77c\ubc18\uc801\uc774\uc9c0\ub9cc, \ud638\uae30\uc2ec\uc5d0 \uac00\ub4dd \ucc2c \ud558\uc774\ud788\ub294 \uc0ac\ub78c\ub4e4\uc774 \ud0c8\ub77d\ud55c \uc21c\uc11c\uac00 \uc8fc\uc5b4\uc9c8 \ub54c $K$\uac00 \ucd5c\uc18c \uba87 \ubc88 \ub2ec\ub77c\uc84c\ub294\uc9c0 \uad6c\ud574\ubcf4\uae30\ub85c \ud588\ub2e4!<\/p>\r\n","input":"<p>\uccab\uc9f8 \uc904\uc5d0\ub294 \ucc38\uac00\uc790\uc758 \uc218 $N$\uacfc \uc815\ud574\uc9c0\ub294 \uc218\uc758 \ucd5c\ub313\uac12 $M$\uc774 \uacf5\ubc31\uc73c\ub85c \uad6c\ubd84\ub418\uc5b4 \uc8fc\uc5b4\uc9c4\ub2e4. $(1\\le N\\le 200\\, 000;$ $1\\le M\\le 10^9)$<\/p>\r\n\r\n<p>\ub458\uc9f8 \uc904\uc5d0\ub294 \ucc38\uac00\uc790\ub4e4\uc758 \ubc88\ud638 $A_1,A_2,\\ldots ,A_N$\uc774 \ud0c8\ub77d\ud55c \uc21c\uc11c\ub300\ub85c \uacf5\ubc31\uc73c\ub85c \uad6c\ubd84\ub418\uc5b4 \uc8fc\uc5b4\uc9c4\ub2e4. $(1\\le A_i\\le N;$ $A_i\\neq A_j\\text{ iff } i\\neq j)$<\/p>\r\n\r\n<p>\uc2e4\uc81c\ub85c \ucc38\uac00\uc790\uac00 \ud574\ub2f9 \uc21c\uc11c\ub300\ub85c \ud0c8\ub77d\ud560 \uc218 \uc788\ub294 \uc785\ub825\ub9cc\uc774 \uc8fc\uc5b4\uc9c4\ub2e4.<\/p>\r\n","output":"<p>\uccab\uc9f8 \uc904\uc5d0 \uac8c\uc784\uc774 \uc9c4\ud589\ub418\uba74\uc11c $K$\uac00 \ucd5c\uc18c \uba87 \ubc88 \ub2ec\ub77c\uc84c\ub294\uc9c0 \ucd9c\ub825\ud55c\ub2e4. $K$\ub97c \ucc98\uc74c \uc815\ud558\ub294 \uac83\uc740 $K$\uac00 \ub2ec\ub77c\uc9c0\ub294 \uac83\uc73c\ub85c \uc138\uc9c0 \uc54a\ub294\ub2e4.<\/p>\r\n","hint":"","original":"1","html_title":"0","problem_lang_tcode":"Korean"},{"problem_id":"34134","problem_lang":"1","title":"[Y] Yet Another Josephus Problem","description":"<p>In the midst of his peaceful life, Hihye was suddenly forced to participate in Bibye&rsquo;s yet another Josephus game! This new Josephus game&rsquo;s rules are as follows:<\/p>\r\n\r\n<ol>\r\n\t<li>Before the game begins, a parameter $M$ is set to a positive integer value.<\/li>\r\n\t<li>The participants sit in a circle in a clockwise order starting from participant number $1$ to participant number $N$.<\/li>\r\n\t<li>When the game starts, the arrow is set to point at participant $N$.<\/li>\r\n\t<li>Then, the following process is repeated until all participants are eliminated.\r\n\t<ul>\r\n\t\t<li>An integer $K$ greater than or equal to $1$ and less than or equal to $M$ is chosen. This value may or may not be differ between iterations.<\/li>\r\n\t\t<li>The arrow is rotated clockwise $K$ times. Eliminated participants&rsquo; seats are skipped.<\/li>\r\n\t\t<li>The participant the arrow points at is eliminated. This action does not result in rotating the arrow.<\/li>\r\n\t<\/ul>\r\n\t<\/li>\r\n<\/ol>\r\n\r\n<p>For example, the following is an example of a possible scenario for this new Josephus game when $N = 7$, and $M = 10$.<\/p>\r\n\r\n<p style=\"text-align: center;\"><img alt=\"\" src=\"https:\/\/upload.acmicpc.net\/e193afc6-3656-4f69-ae24-6d5ed08830ba\/-\/preview\/\" style=\"width: 797px; max-width: 100%\" \/><\/p>\r\n\r\n<p>Normally, one would be curious where they should be in order to survive till the end. However, Hihye wants to find out the minimum number of times $K$ could have been changed between iterations given the order in which the players were eliminated!<\/p>\r\n","input":"<p>The first line of input contains the number of participants $N$, and the maximum value which can be chosen $M$, separated by a space. $(1\\le N\\le 200\\, 000;$ $1\\le M\\le 10^9)$<\/p>\r\n\r\n<p>The second line contains the numbers belonging to the participants $A_1,A_2,\\ldots ,A_N$ in the order the participants were eliminated. $(1\\le A_i\\le N;$ $A_i\\neq A_j\\text{ iff } i\\neq j)$<\/p>\r\n\r\n<p>Only input data that allows participants to be eliminated in the given order is provided.<\/p>\r\n","output":"<p>Print the minimum number of times $K$ could have been changed between iterations. Choosing $K$ for the first time when the game starts does not count as $K$ changing.<\/p>\r\n","hint":"","original":"0","html_title":"0","problem_lang_tcode":"English"}]