A palindrome must be mirrored over the center. Suppose we have a palindrome. If we prepend the word "ab" on the left, what must we append on the right to keep it a palindrome?

We must append "ba" on the right. The number of times we can do this is the minimum of (occurrences of "ab") and (occurrences of "ba").

For words that are already palindromes, e.g. "aa", we can prepend and append these in pairs as described in the previous hint. We can also use exactly one in the middle to form an even longer palindrome.