
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 \( n \geqslant 3 \) be a positive integer. A sequence of distinct points \( P_{1}, P_{2}, \cdots, P_{n} \) in the plane is called "good" if no three points are collinear, the polygonal path \( P_{1} P_{2} \cdots P_{n} \) (formed by the segments \( P_{1} P_{2}, P_{2} P_{3}, \cdots, P_{n-1} P_{n} \)) is non-self-intersecting (i.e., no two segments \( P_{1} P_{2}, P_{2} P_{3}, \cdots, P_{n-1} P_{n} \) share a common point other than \( P_{1}, P_{2}, \cdots, P_{n} \)), and all points \( P_{i}, P_{i+1}, P_{i+2} \) for \( i=1, 2, \cdots, n-2 \) are arranged counterclockwise in increasing order of their indices.

For each positive integer \( n (n \geqslant 3) \), find the largest integer \( f(n) \) such that there exist \( n \) distinct points \( A_{1}, A_{2}, \cdots, A_{n} \) in the plane, and there are \( f(n) \) distinct permutations
\[ 
\sigma : \{1, 2, \cdots, n\} \rightarrow \{1, 2, \cdots, n\} 
\]
such that the sequence \( A_{\sigma(1)}, A_{\sigma(2)}, \cdots, A_{\sigma(n)} \) is good.

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