문제
때는 2123년, UDPC는 전세계의 많은 지역을 아우르는 거대한 대회로 자라났다. UDPC를 구성하는 $N$개의 지역은 UDPC의 100주년을 맞아 모든 지역을 통일해 연합을 구성하기로 했다. 각 지역은 1ドル$부터 $N$까지의 지역 번호로 구분된다. 하지만 통일 과정에서 사소한 문제가 발생하는데, 바로 지역마다 지역 코드가 제각각이라는 점이다.
지역 코드는 지역 번호와 다른 것으로, 각 지역에 배정된 0ドル$부터 9ドル$까지의 숫자로 이루어진 문자열이다. 긴 회의 끝에, 다음과 같은 규칙을 통해 새 연합의 지역 코드를 결정하기로 하였다.
- 새 지역 코드는 기존 지역의 지역 코드 중 하나 이상을 골라 합친 뒤 문자 단위로 재배열한 것이어야 한다.
- 새 지역 코드는 외우기 쉽도록 회문이어야 한다. 회문이란, 왼쪽에서 오른쪽으로 읽은 것과 오른쪽에서 왼쪽으로 읽은 것이 서로 같은 문자열을 말한다.
- 새 지역 코드는 위의 조건을 만족하는 것 중 가장 짧은 것이다.
UDPC의 평화를 위해, 새 연합의 지역 코드를 구해 보자.
출력
첫째 줄에 새 지역 코드의 길이를 출력한다. 만약 새 지역 코드를 만드는 것이 불가능할 경우 -1을 출력한다.
새 지역 코드를 만드는 것이 가능할 경우 아래와 같이 두 줄을 더 출력한다.
둘째 줄에 새 지역 코드를 이루는 지역의 수 $M$을 출력한다.
셋째 줄에 새 지역 코드를 이루는 $M$개 지역의 지역 번호를 공백으로 구분해 오름차순으로 출력한다.
가능한 방법이 여러 가지일 경우 그중 아무 것이나 출력한다.
[{"problem_id":"33698","problem_lang":"0","title":"\uc9c0\uc5ed \ucf54\ub4dc \uc815\ud558\uae30","description":"<p>\ub54c\ub294 2123\ub144, UDPC\ub294 \uc804\uc138\uacc4\uc758 \ub9ce\uc740 \uc9c0\uc5ed\uc744 \uc544\uc6b0\ub974\ub294 \uac70\ub300\ud55c \ub300\ud68c\ub85c \uc790\ub77c\ub0ac\ub2e4. UDPC\ub97c \uad6c\uc131\ud558\ub294 $N$\uac1c\uc758 \uc9c0\uc5ed\uc740 UDPC\uc758 100\uc8fc\ub144\uc744 \ub9de\uc544 \ubaa8\ub4e0 \uc9c0\uc5ed\uc744 \ud1b5\uc77c\ud574 \uc5f0\ud569\uc744 \uad6c\uc131\ud558\uae30\ub85c \ud588\ub2e4. \uac01 \uc9c0\uc5ed\uc740 $1$\ubd80\ud130 $N$\uae4c\uc9c0\uc758 <strong>\uc9c0\uc5ed \ubc88\ud638<\/strong>\ub85c \uad6c\ubd84\ub41c\ub2e4. \ud558\uc9c0\ub9cc \ud1b5\uc77c \uacfc\uc815\uc5d0\uc11c \uc0ac\uc18c\ud55c \ubb38\uc81c\uac00 \ubc1c\uc0dd\ud558\ub294\ub370, \ubc14\ub85c \uc9c0\uc5ed\ub9c8\ub2e4 <strong>\uc9c0\uc5ed \ucf54\ub4dc<\/strong>\uac00 \uc81c\uac01\uac01\uc774\ub77c\ub294 \uc810\uc774\ub2e4.<\/p>\r\n\r\n<p><strong>\uc9c0\uc5ed \ucf54\ub4dc<\/strong>\ub294 <strong>\uc9c0\uc5ed \ubc88\ud638<\/strong>\uc640 \ub2e4\ub978 \uac83\uc73c\ub85c, \uac01 \uc9c0\uc5ed\uc5d0 \ubc30\uc815\ub41c $0$\ubd80\ud130 $9$\uae4c\uc9c0\uc758 \uc22b\uc790\ub85c \uc774\ub8e8\uc5b4\uc9c4 \ubb38\uc790\uc5f4\uc774\ub2e4. \uae34 \ud68c\uc758 \ub05d\uc5d0, \ub2e4\uc74c\uacfc \uac19\uc740 \uaddc\uce59\uc744 \ud1b5\ud574 \uc0c8 \uc5f0\ud569\uc758 \uc9c0\uc5ed \ucf54\ub4dc\ub97c \uacb0\uc815\ud558\uae30\ub85c \ud558\uc600\ub2e4.<\/p>\r\n\r\n<ul>\r\n\t<li>\uc0c8 \uc9c0\uc5ed \ucf54\ub4dc\ub294 \uae30\uc874 \uc9c0\uc5ed\uc758 \uc9c0\uc5ed \ucf54\ub4dc \uc911 \ud558\ub098 \uc774\uc0c1\uc744 \uace8\ub77c \ud569\uce5c \ub4a4 \ubb38\uc790 \ub2e8\uc704\ub85c \uc7ac\ubc30\uc5f4\ud55c \uac83\uc774\uc5b4\uc57c \ud55c\ub2e4.<\/li>\r\n\t<li>\uc0c8 \uc9c0\uc5ed \ucf54\ub4dc\ub294 \uc678\uc6b0\uae30 \uc27d\ub3c4\ub85d <strong>\ud68c\ubb38<\/strong>\uc774\uc5b4\uc57c \ud55c\ub2e4. \ud68c\ubb38\uc774\ub780, \uc67c\ucabd\uc5d0\uc11c \uc624\ub978\ucabd\uc73c\ub85c \uc77d\uc740 \uac83\uacfc \uc624\ub978\ucabd\uc5d0\uc11c \uc67c\ucabd\uc73c\ub85c \uc77d\uc740 \uac83\uc774 \uc11c\ub85c \uac19\uc740 \ubb38\uc790\uc5f4\uc744 \ub9d0\ud55c\ub2e4.<\/li>\r\n\t<li>\uc0c8 \uc9c0\uc5ed \ucf54\ub4dc\ub294 \uc704\uc758 \uc870\uac74\uc744 \ub9cc\uc871\ud558\ub294 \uac83 \uc911 \uac00\uc7a5 \uc9e7\uc740 \uac83\uc774\ub2e4.<\/li>\r\n<\/ul>\r\n\r\n<p>UDPC\uc758 \ud3c9\ud654\ub97c \uc704\ud574, \uc0c8 \uc5f0\ud569\uc758 \uc9c0\uc5ed \ucf54\ub4dc\ub97c \uad6c\ud574 \ubcf4\uc790.<\/p>\r\n","input":"<p>\uccab\uc9f8 \uc904\uc5d0 \uc9c0\uc5ed\uc758 \uc218 $N$\uc774 \uc8fc\uc5b4\uc9c4\ub2e4. $(1\\le N\\le 10^6)$<\/p>\r\n\r\n<p>\ub458\uc9f8 \uc904\ubd80\ud130 $N$\uac1c\uc758 \uc904\uc774 \uc8fc\uc5b4\uc9c4\ub2e4. \uadf8\uc911 $i$\ubc88\uc9f8 \uc904\uc5d0 $i$\ubc88 \uc9c0\uc5ed\uc758 \uc9c0\uc5ed \ucf54\ub4dc\ub97c \ub098\ud0c0\ub0b4\ub294 \ube44\uc5b4 \uc788\uc9c0 \uc54a\uc740 \ubb38\uc790\uc5f4 $S_i$\uac00 \uc8fc\uc5b4\uc9c4\ub2e4.<\/p>\r\n\r\n<p>\uac01 \ubb38\uc790\uc5f4\uc740 $0$\ubd80\ud130 $9$\uae4c\uc9c0\uc758 \uc22b\uc790\ub85c\ub9cc \uc774\ub8e8\uc5b4\uc838 \uc788\uc73c\uba70, \ubaa8\ub4e0 \ubb38\uc790\uc5f4\uc758 \uae38\uc774\uc758 \ud569\uc740 $2\\times 10^6$ \uc774\ud558\uc784\uc774 \ubcf4\uc7a5\ub41c\ub2e4.<\/p>\r\n","output":"<p>\uccab\uc9f8 \uc904\uc5d0 \uc0c8 \uc9c0\uc5ed \ucf54\ub4dc\uc758 \uae38\uc774\ub97c \ucd9c\ub825\ud55c\ub2e4. \ub9cc\uc57d \uc0c8 \uc9c0\uc5ed \ucf54\ub4dc\ub97c \ub9cc\ub4dc\ub294 \uac83\uc774 \ubd88\uac00\ub2a5\ud560 \uacbd\uc6b0 <span style=\"color:#e74c3c;\"><code>-1<\/code><\/span>\uc744 \ucd9c\ub825\ud55c\ub2e4.<\/p>\r\n\r\n<p>\uc0c8 \uc9c0\uc5ed \ucf54\ub4dc\ub97c \ub9cc\ub4dc\ub294 \uac83\uc774 \uac00\ub2a5\ud560 \uacbd\uc6b0 \uc544\ub798\uc640 \uac19\uc774 \ub450 \uc904\uc744 \ub354 \ucd9c\ub825\ud55c\ub2e4.<\/p>\r\n\r\n<p>\ub458\uc9f8 \uc904\uc5d0 \uc0c8 \uc9c0\uc5ed \ucf54\ub4dc\ub97c \uc774\ub8e8\ub294 \uc9c0\uc5ed\uc758 \uc218 $M$\uc744 \ucd9c\ub825\ud55c\ub2e4.<\/p>\r\n\r\n<p>\uc14b\uc9f8 \uc904\uc5d0 \uc0c8 \uc9c0\uc5ed \ucf54\ub4dc\ub97c \uc774\ub8e8\ub294 $M$\uac1c \uc9c0\uc5ed\uc758 <strong>\uc9c0\uc5ed \ubc88\ud638<\/strong>\ub97c \uacf5\ubc31\uc73c\ub85c \uad6c\ubd84\ud574 \uc624\ub984\ucc28\uc21c\uc73c\ub85c \ucd9c\ub825\ud55c\ub2e4.<\/p>\r\n\r\n<p>\uac00\ub2a5\ud55c \ubc29\ubc95\uc774 \uc5ec\ub7ec \uac00\uc9c0\uc77c \uacbd\uc6b0 \uadf8\uc911 \uc544\ubb34 \uac83\uc774\ub098 \ucd9c\ub825\ud55c\ub2e4.<\/p>\r\n","hint":"","original":"1","html_title":"0","problem_lang_tcode":"Korean"},{"problem_id":"33698","problem_lang":"1","title":"Region Code","description":"<p>In the year 2123, UDPC has grown into a massive competition spanning many regions worldwide. To celebrate the 100th anniversary of UDPC, the $N$ regions that make up UDPC have decided to unite into a single federation. Each region is identified by a <strong>region number<\/strong> ranging from $1$ to $N$. However, a minor issue has arisen during the unification process: the fact that&nbsp;<strong>region codes<\/strong> vary by region.<\/p>\r\n\r\n<p>A <strong>region code<\/strong> is distinct from a <strong>region number<\/strong>; it is a string consisting of digits from $0$ to $9$ assigned to each region. After a long discussion, the following rules were established to determine the new federation&rsquo;s region code:<\/p>\r\n\r\n<ul>\r\n\t<li>The new region code must be formed by selecting one or more of the existing region codes and then rearranging their characters.<\/li>\r\n\t<li>The new region code must be a <strong>palindrome<\/strong> to make it easier to remember. A palindrome is a string that reads the same backward as forward.<\/li>\r\n\t<li>Among all valid options, the new region code must be the shortest possible.<\/li>\r\n<\/ul>\r\n\r\n<p>For the sake of UDPC&rsquo;s unity, determine the new federation&rsquo;s region code.<\/p>\r\n","input":"<p>The first line of input contains an integer $N$, representing the number of regions. $(1 \\le N \\le 10^6)$<\/p>\r\n\r\n<p>The $i$-th of the following $N$ lines contains a nonempty string $S_i$, the region code of the $i$-th region.<\/p>\r\n\r\n<p>Each string consists only of digits from $0$ to $9$, and it is guaranteed the total length of all strings does not exceed $2 \\times 10^6$.<\/p>\r\n","output":"<p>On the first line, print the length of the new region code. If it is impossible to create a new valid region code, print <span style=\"color:#e74c3c;\"><code>-1<\/code><\/span>.<\/p>\r\n\r\n<p>If a new valid region code can be formed, print two more lines as below.<\/p>\r\n\r\n<p>On the second line, print an integer $M$, the number of regions used to construct the new region code.<\/p>\r\n\r\n<p>On the third line, print $M$ integers in ascending order, the <strong>region numbers<\/strong>&nbsp;of these $M$ regions.<\/p>\r\n\r\n<p>If there are multiple possible solutions, print any of them.<\/p>\r\n","hint":"","original":"0","html_title":"0","problem_lang_tcode":"English"}]