Kira Loves Palindromes

You are given a string $S$. Find the number of ways to choose an unordered pair of non-overlapping non-empty substrings of this string (let's denote them by $s_1$ and $s_2$ in such a way that $s_2$ starts after $s_1$ ends) such that their concatenation $s_1 + s_2$ is a palindrome. Two pairs $(s_1, s_2)$ and $(s_1', s_2')$ are different if $s_1$ is chosen at a different position from $s_1'$ or $s_