Circular Logic II

Given an integer $n$ , $n \geq 3$ , let $B = {false, true}$ and let $B^{n}$ be the set of sequences of $n$ values from $B$ . The function $f$ from $B^{n}$ to $B^{n}$ is defined by $f (b_{1} \dots b_{n}) = c_{1} \dots c_{n}$ where:

$c_{i} = b_{i + 1}$ for $1 \leq i < n$ .
$c_{n} = b_{1} AND (b_{2} XOR b_{3})$ , where $AND$ and $XOR$ are the logical $AND$ and exclusive $OR$ operations.

Let $S (n)$ be the number of functions $T$ from $B^{n}$ to $B$ such that for all $x$ in $B^{n}$ , $T (x) AND T (f (x)) = false$ . You are given that $S (3) = 35$ and $S (4) = 2118$ .

Find $S (20)$ . Give your answer modulo $1 001 001 011$ .

Write-up coming later

The complete problem is available here. An approach, code, and answer will be added later.