Maximum Value (Hard Version)

You are provided with a binary string $S$ of length $N$. You have to choose a substring $G$ from $S$ having even length and divide it into the two equal parts $P1$ and $P2$ such that $P1$ will be the first half of $G$ and $P2$ will be the second. For some reason, reverse the part $P2$. There is a function $F(X)$, such that: $F(X)$ = $2*Len(P1) / (1 +$ $2^{|X|}$ $)$ Where: $X$ = Binary string for