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문제

벌은 가장 근면성실한 곤충들 중의 하나입니다. 꽃에서 꿀과 꽃가루를 수집하기위해, 벌들은 숲 속의 나무들을 의지할 수 밖에 없습니다. 벌들은 작업을 단순하게 하기 위하여 n 개의 나무에 0 부터 n - 1 까지의 번호를 붙였습니다. 숲 전체를 돌아다니는 대신에, 경로들을 모아놓은 특정한 목록을 이용합니다. 하나의 경로는 두 개의 나무로 구성되고, 벌들은 어느 방향으로든 이동할 수 있습니다 (하나의 나무에서 다른 나무까지 직선으로). 그 목록에 없는 경로는 이용하지 않습니다.

기술이 꽤 발전하면서, 벌들은 일하는 전략 역시 변경했습니다. 숲 속에 있는 모든 나무들 위를 맴도는 대신에, 꽃들이 많은 특정 나무들을 목표로 삼고자 합니다. 그래서, 벌들은 목표한 몇몇 나무들에 새로운 벌집을 건설할 계획을 세웠습니다. 벌집이 모두 완성되면, 벌집이 있는 나무들에서만 식량을 모으려고 합니다. 그리고 벌집이 없는 나무에는 가지 않도록 하기 위해서 몇몇 경로들을 목록에서 지울 것 입니다. 

자 이제, 벌들은 벌집을 건설하려고 합니다. 이 벌집들은 여러 경로 중 하나에 문제가 발생하더라도 (해당 경로 상에서 새나 동물들이 벌들을 방해할 수 있습니다.) 존재하는 다른 경로들을 이용해서 모든 벌집 간의 이동이 가능해야 합니다.

벌집 하나를 완성하는데는 많은 노력이 들기 때문에 벌들은 ,적어도 두 개 이상의 나무에, 가능한 적은 수의 벌집을 유지하고자 합니다. 이제 벌들이 이용하는 여러 경로와 나무를 이용해서, 새로운 벌집 군집를 제안해 봅시다.

입력

입력은 테스트 케이스들의 수를 나타내는 정수 T (T ≤ 50) 와 시작하고, 

각각의 케이스는 하나의 빈 줄로 시작합니다. 다음 라인부터 두 개의 정수 n (2 ≤ n ≤ 500) 과 m (0 ≤ m ≤ 20000) 이 주어지고, n 은 나무들의 개수를 나타내고, m 은 경로들의 개수를 나타냅니다. 다음 m 라인은 두 개의 정수 u v (0 ≤ u, v < n, u ≠ v) 이고, 나무 u 에서 나무 v 로의 하나의 경로를 의미합니다. 나무 u 에서 나무 v 의 경로는 하나의 경로만 있다고 가정합니다. 말할 필요가 없긴 하지만, 입력으로 주어진 하나의 경로는 다시 주어지지 않을 것입니다.

출력

각각의 경우에 대해서, 문제 번호와 제안된 벌집 군집에서의 벌집의 개수를 출력하거나, 그러한 군집을 찾을 수 없다면 ‘impossible’을 출력합니다.

예제 입력 1

3

3 3
0 1
1 2
2 0

2 1
0 1

5 6
0 1
1 2
1 3
2 3
0 4
3 4

예제 출력 1

Case 1: 3
Case 2: impossible
Case 3: 3

힌트

입력데이터가 큽니다. 빠른 I/O 처리를 이용하세요.

[{"problem_id":"3805","problem_lang":"0","title":"\ubc8c\uc9d1\ub4e4","description":"<p>\ubc8c\uc740 \uac00\uc7a5 \uadfc\uba74\uc131\uc2e4\ud55c&nbsp;\uace4\ucda9\ub4e4 \uc911\uc758 \ud558\ub098\uc785\ub2c8\ub2e4.&nbsp;\uaf43\uc5d0\uc11c&nbsp;\uafc0\uacfc \uaf43\uac00\ub8e8\ub97c \uc218\uc9d1\ud558\uae30\uc704\ud574, \ubc8c\ub4e4\uc740 \uc232 \uc18d\uc758 \ub098\ubb34\ub4e4\uc744 \uc758\uc9c0\ud560 \uc218 \ubc16\uc5d0 \uc5c6\uc2b5\ub2c8\ub2e4.&nbsp;\ubc8c\ub4e4\uc740 \uc791\uc5c5\uc744 \ub2e8\uc21c\ud558\uac8c \ud558\uae30 \uc704\ud558\uc5ec n \uac1c\uc758 \ub098\ubb34\uc5d0 0 \ubd80\ud130 n - 1 \uae4c\uc9c0\uc758 \ubc88\ud638\ub97c \ubd99\uc600\uc2b5\ub2c8\ub2e4. \uc232 \uc804\uccb4\ub97c \ub3cc\uc544\ub2e4\ub2c8\ub294 \ub300\uc2e0\uc5d0,&nbsp;\uacbd\ub85c\ub4e4\uc744 \ubaa8\uc544\ub193\uc740 \ud2b9\uc815\ud55c&nbsp;\ubaa9\ub85d\uc744 \uc774\uc6a9\ud569\ub2c8\ub2e4. \ud558\ub098\uc758 \uacbd\ub85c\ub294 \ub450 \uac1c\uc758 \ub098\ubb34\ub85c \uad6c\uc131\ub418\uace0, \ubc8c\ub4e4\uc740 \uc5b4\ub290 \ubc29\ud5a5\uc73c\ub85c\ub4e0&nbsp;\uc774\ub3d9\ud560 \uc218 \uc788\uc2b5\ub2c8\ub2e4 (\ud558\ub098\uc758 \ub098\ubb34\uc5d0\uc11c \ub2e4\ub978 \ub098\ubb34\uae4c\uc9c0 \uc9c1\uc120\uc73c\ub85c).&nbsp;\uadf8 \ubaa9\ub85d\uc5d0 \uc5c6\ub294 \uacbd\ub85c\ub294 \uc774\uc6a9\ud558\uc9c0 \uc54a\uc2b5\ub2c8\ub2e4.<\/p>\r\n\r\n<p>\uae30\uc220\uc774 \uaf64 \ubc1c\uc804\ud558\uba74\uc11c, \ubc8c\ub4e4\uc740&nbsp;\uc77c\ud558\ub294 \uc804\ub7b5 \uc5ed\uc2dc&nbsp;\ubcc0\uacbd\ud588\uc2b5\ub2c8\ub2e4. \uc232 \uc18d\uc5d0 \uc788\ub294 \ubaa8\ub4e0 \ub098\ubb34\ub4e4 \uc704\ub97c \ub9f4\ub3c4\ub294 \ub300\uc2e0\uc5d0, \uaf43\ub4e4\uc774 \ub9ce\uc740 \ud2b9\uc815 \ub098\ubb34\ub4e4\uc744 \ubaa9\ud45c\ub85c \uc0bc\uace0\uc790 \ud569\ub2c8\ub2e4. \uadf8\ub798\uc11c, \ubc8c\ub4e4\uc740&nbsp;\ubaa9\ud45c\ud55c \uba87\uba87 \ub098\ubb34\ub4e4\uc5d0&nbsp;\uc0c8\ub85c\uc6b4 \ubc8c\uc9d1\uc744 \uac74\uc124\ud560 \uacc4\ud68d\uc744 \uc138\uc6e0\uc2b5\ub2c8\ub2e4. \ubc8c\uc9d1\uc774 \ubaa8\ub450 \uc644\uc131\ub418\uba74,&nbsp;\ubc8c\uc9d1\uc774 \uc788\ub294&nbsp;\ub098\ubb34\ub4e4\uc5d0\uc11c\ub9cc&nbsp;\uc2dd\ub7c9\uc744 \ubaa8\uc73c\ub824\uace0&nbsp;\ud569\ub2c8\ub2e4. \uadf8\ub9ac\uace0&nbsp;\ubc8c\uc9d1\uc774 \uc5c6\ub294&nbsp;\ub098\ubb34\uc5d0\ub294 \uac00\uc9c0 \uc54a\ub3c4\ub85d \ud558\uae30 \uc704\ud574\uc11c \uba87\uba87 \uacbd\ub85c\ub4e4\uc744 \ubaa9\ub85d\uc5d0\uc11c \uc9c0\uc6b8 \uac83 \uc785\ub2c8\ub2e4.&nbsp;<\/p>\r\n\r\n<p>\uc790 \uc774\uc81c, \ubc8c\ub4e4\uc740 \ubc8c\uc9d1\uc744 \uac74\uc124\ud558\ub824\uace0 \ud569\ub2c8\ub2e4. \uc774 \ubc8c\uc9d1\ub4e4\uc740 \uc5ec\ub7ec \uacbd\ub85c&nbsp;\uc911&nbsp;\ud558\ub098\uc5d0&nbsp;\ubb38\uc81c\uac00 \ubc1c\uc0dd\ud558\ub354\ub77c\ub3c4 (\ud574\ub2f9&nbsp;\uacbd\ub85c \uc0c1\uc5d0\uc11c \uc0c8\ub098 \ub3d9\ubb3c\ub4e4\uc774 \ubc8c\ub4e4\uc744 \ubc29\ud574\ud560 \uc218 \uc788\uc2b5\ub2c8\ub2e4.) \uc874\uc7ac\ud558\ub294 \ub2e4\ub978 \uacbd\ub85c\ub4e4\uc744 \uc774\uc6a9\ud574\uc11c \ubaa8\ub4e0&nbsp;\ubc8c\uc9d1 \uac04\uc758&nbsp;\uc774\ub3d9\uc774 \uac00\ub2a5\ud574\uc57c \ud569\ub2c8\ub2e4.<\/p>\r\n\r\n<p>\ubc8c\uc9d1&nbsp;\ud558\ub098\ub97c \uc644\uc131\ud558\ub294\ub370\ub294&nbsp;\ub9ce\uc740 \ub178\ub825\uc774 \ub4e4\uae30 \ub54c\ubb38\uc5d0&nbsp;\ubc8c\ub4e4\uc740 ,\uc801\uc5b4\ub3c4 \ub450 \uac1c \uc774\uc0c1\uc758 \ub098\ubb34\uc5d0,&nbsp;\uac00\ub2a5\ud55c \uc801\uc740 \uc218\uc758 \ubc8c\uc9d1\uc744 \uc720\uc9c0\ud558\uace0\uc790 \ud569\ub2c8\ub2e4.&nbsp;\uc774\uc81c \ubc8c\ub4e4\uc774 \uc774\uc6a9\ud558\ub294 \uc5ec\ub7ec \uacbd\ub85c\uc640&nbsp;\ub098\ubb34\ub97c&nbsp;\uc774\uc6a9\ud574\uc11c,&nbsp;\uc0c8\ub85c\uc6b4 \ubc8c\uc9d1 \uad70\uc9d1\ub97c \uc81c\uc548\ud574 \ubd05\uc2dc\ub2e4.<\/p>\r\n","input":"<p>\uc785\ub825\uc740 \ud14c\uc2a4\ud2b8 \ucf00\uc774\uc2a4\ub4e4\uc758 \uc218\ub97c \ub098\ud0c0\ub0b4\ub294 \uc815\uc218 T (T &le; 50) \uc640 \uc2dc\uc791\ud558\uace0,&nbsp;<\/p>\r\n\r\n<p>\uac01\uac01\uc758 \ucf00\uc774\uc2a4\ub294 \ud558\ub098\uc758 \ube48 \uc904\ub85c \uc2dc\uc791\ud569\ub2c8\ub2e4. \ub2e4\uc74c \ub77c\uc778\ubd80\ud130 \ub450 \uac1c\uc758 \uc815\uc218 n (2 &le; n &le; 500) \uacfc m (0 &le; m &le; 20000) \uc774 \uc8fc\uc5b4\uc9c0\uace0, n \uc740 \ub098\ubb34\ub4e4\uc758 \uac1c\uc218\ub97c \ub098\ud0c0\ub0b4\uace0, m \uc740 \uacbd\ub85c\ub4e4\uc758 \uac1c\uc218\ub97c \ub098\ud0c0\ub0c5\ub2c8\ub2e4. \ub2e4\uc74c m \ub77c\uc778\uc740 \ub450 \uac1c\uc758 \uc815\uc218 u v (0 &le; u, v &lt; n, u &ne; v) \uc774\uace0, \ub098\ubb34 u \uc5d0\uc11c \ub098\ubb34 v \ub85c\uc758 \ud558\ub098\uc758 \uacbd\ub85c\ub97c \uc758\ubbf8\ud569\ub2c8\ub2e4. \ub098\ubb34 u \uc5d0\uc11c \ub098\ubb34 v \uc758 \uacbd\ub85c\ub294 \ud558\ub098\uc758 \uacbd\ub85c\ub9cc \uc788\ub2e4\uace0 \uac00\uc815\ud569\ub2c8\ub2e4. \ub9d0\ud560 \ud544\uc694\uac00&nbsp;\uc5c6\uae34 \ud558\uc9c0\ub9cc, \uc785\ub825\uc73c\ub85c \uc8fc\uc5b4\uc9c4 \ud558\ub098\uc758 \uacbd\ub85c\ub294 \ub2e4\uc2dc&nbsp;\uc8fc\uc5b4\uc9c0\uc9c0 \uc54a\uc744 \uac83\uc785\ub2c8\ub2e4.<\/p>\r\n","output":"<p>\uac01\uac01\uc758 \uacbd\uc6b0\uc5d0 \ub300\ud574\uc11c, \ubb38\uc81c&nbsp;\ubc88\ud638\uc640 \uc81c\uc548\ub41c \ubc8c\uc9d1 \uad70\uc9d1\uc5d0\uc11c\uc758 \ubc8c\uc9d1\uc758 \uac1c\uc218\ub97c \ucd9c\ub825\ud558\uac70\ub098, \uadf8\ub7ec\ud55c \uad70\uc9d1\uc744&nbsp;\ucc3e\uc744 \uc218 \uc5c6\ub2e4\uba74 &lsquo;impossible&rsquo;\uc744 \ucd9c\ub825\ud569\ub2c8\ub2e4.<\/p>\r\n","hint":"<p>\uc785\ub825\ub370\uc774\ud130\uac00 \ud07d\ub2c8\ub2e4. \ube60\ub978 I\/O \ucc98\ub9ac\ub97c \uc774\uc6a9\ud558\uc138\uc694.<\/p>\r\n","original":"0","problem_lang_code":"\ud55c\uad6d\uc5b4"},{"problem_id":"3805","problem_lang":"1","title":"Beehives","description":"<p>Bees are one of the most industrious insects. Since they collect nectar and pollen from flowers, they have to rely on the trees in the forest. For simplicity they numbered the n trees from 0 to n &minus; 1. Instead of roaming around all over the forest, they use a particular list of paths. A path is based on two trees, and they can move either way i.e. from one tree to another in straight line. They don&rsquo;t use paths that are not in their list.<\/p>\r\n\r\n<p>As technology has been improved a lot, they also changed their working strategy. Instead of hovering over all the trees in the forest, they are targeting particular trees, mainly trees with lots of flowers. So, they planned that they will build some new hives in some targeted trees. After that they will only collect their foods from these trees. They will also remove some paths from their list so that they don&rsquo;t have to go to a tree with no hive in it.<\/p>\r\n\r\n<p>Now, they want to build the hives such that if one of the paths in their new list go down (some birds or animals disturbs them in that path) it&rsquo;s still possible to go from any hive to another using the existing paths.<\/p>\r\n\r\n<p>They don&rsquo;t want to choose less than two trees and as hive-building requires a lot of work, they need to keep the number of hives as low as possible. Now you are given the trees with the paths they use, your task is to propose a new bee hive colony for them.<\/p>\r\n","input":"<p>Input starts with an integer T (T &le; 50), denoting the number of test cases.<\/p>\r\n\r\n<p>Each case starts with a blank line. Next line contains two integers n (2 &le; n &le; 500) and m (0 &le; m &le; 20000), where n denotes the number of trees and m denotes the number of paths. Each of the next m lines contains two integers u v (0 &le; u, v &lt; n, u &ne; v) meaning that there is a path between tree u and v. Assume that there can be at most one path between tree u to v, and needless to say that a path will not be given more than once in the input.<\/p>\r\n","output":"<p>For each case, print the case number and the number of beehives in the proposed colony or &lsquo;impossible&rsquo; if its impossible to find such a colony.<\/p>\r\n","hint":"<p>Dataset is huge. Use faster I\/O methods.<\/p>\r\n","original":"1","problem_lang_code":"\uc601\uc5b4"}]