Palindrome Free Strings

You are given a string $S$ consisting of characters 0, 1, and ?. You can replace each ? with either 0 or 1. Your task is to find if it is possible to assign each ? to either 0 or 1 such that the resulting string has no substrings that are palindromes of length $5$ or more.

The first line of the input gives the number of test cases, $T$. $T$ test cases follow.

Each test case consists of two lines.

The first line of each test case contains an integer $N$, denoting the length of the string $S$.

The second line of each test case contains a string $S$ of length $N$.

For each test case, output one line containing Case #x: y, where $x$ is the test case number (starting from 1) and $y$ is POSSIBLE if there is a possible resulting string that has no palindromic substrings of length $5$ or more, or IMPOSSIBLE otherwise.

• $1≤T≤100$.
• $S$ only consists of characters 0, 1 and ?.

In Sample Case #1, to prevent the whole string from being a palindrome, the first and last question mark must be different characters.

If we replace first question mark with 0 and replace the last question mark with 1, we get 1000?1001. If the remaining ? is replaced by 1, we get 100011001, then the first $5$ characters form a palindrome of length $5$. Otherwise, we get 100001001, the first $6$ characters are a palindrome of length $6$.

If we replace first question mark with 1 we get 1001?0001. If the remaining ? is replaced by 1, we get 100110001, then the last $5$ characters form a palindrome of length $5$. Otherwise, we get 100100001, the last $6$ characters are a palindrome of length $6$.

Hence, there is no way to get a valid string.

In Sample Case #2, one of the valid strings after replacing all the ? is 10011.