## Unpredictable Permutations

### Problem 720

Consider all permutations of $\{1, 2, \ldots N\}$, listed in lexicographic order.

For example, for $N=4$, the list starts as follows:
\[
(1, 2, 3, 4) \\
(1, 2, 4, 3) \\
(1, 3, 2, 4) \\
(1, 3, 4, 2) \\
(1, 4, 2, 3) \\
(1, 4, 3, 2) \\
(2, 1, 3, 4) \\
\vdots
\]

Let us call a permutation $P$ *unpredictable* if there is no choice of three indices $i \lt j \lt k$ such that $P(i)$, $P(j)$ and $P(k)$ constitute an arithmetic progression.

For example, $P=(3, 4, 2, 1)$ is *not* unpredictable because $P(1), P(3), P(4)$ is an arithmetic progression.

Let $S(N)$ be the position within the list of the first unpredictable permutation.

For example, given $N = 4$, the first unpredictable permutation is $(1, 3, 2, 4)$ so $S(4) = 3$.

You are also given that $S(8) = 2295$ and $S(32) \equiv 641839205 \pmod{1\,000\,000\,007}$.

Find $S(2^{25})$. Give your answer modulo $1\,000\,000\,007$.