
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
Let \( S = \left\{ (x, y) \in \mathbb{Z}^{2} \mid 0 \leq x \leq 11, 0 \leq y \leq 9 \right\} \). Compute the number of sequences \(\left( s_{0}, s_{1}, \ldots, s_{n} \right)\) of elements in \( S \) (for any positive integer \( n \geq 2 \)) that satisfy the following conditions:
- \( s_{0} = (0,0) \) and \( s_{1} = (1,0) \),
- \( s_{0}, s_{1}, \ldots, s_{n} \) are distinct,
- For all integers \( 2 \leq i \leq n \), \( s_{i} \) is obtained by rotating \( s_{i-2} \) about \( s_{i-1} \) by either \( 90^{\circ} \) or \( 180^{\circ} \) in the clockwise direction.

Present the answer in LaTex format: \boxed{Your answer}
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