Making a Pelindrome

Let the completed string be $X$.

If the prefix $X[1..i]$ of length $i$ is a palindrome, its first and last characters must be equal. Therefore, $X_1=X_i$ for every $i$ ($1 \leq i \leq N$). Thus, every character of $X$ must be the same.

It remains to inspect the uppercase letters already fixed in $S$.

• If there is no fixed uppercase letter, any of the 26 uppercase letters can be the common character, so the answer is $26$.
• If all fixed uppercase letters are equal, the common character is uniquely determined, so the answer is $1$.
• If two different fixed uppercase letters exist, making all characters equal is impossible, so the answer is $0$.

Scanning each test case once is sufficient. The time complexity is $O(N)$, and the extra space complexity is $O(1)$.