Let $S = \{(x, y) \in Z^2 | 0 \le x \le 11, 0\le y \le 9\}$. Compute the number of sequences $(s_0, s_1, . . . , s_n)$ of elements in $S$ (for any positive integer $n \ge 2$) that satisfy the following conditions:
$\bullet$ $s_0 = (0, 0)$ and $s_1 = (1, 0)$,
$\bullet$ $s_0, s_1, . . . , s_n$ are distinct,
$\bullet$ for all integers $2  \le  i  \le  n$, $s_i$ is obtained by rotating $s_{i-2}$ about $s_{i-1}$ by either $90^o$ or $180^o$ in the
clockwise direction.

Let's think step by step and output the final answer within \boxed{}.
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