
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  $n \geq 3$  be an integer. A sequence  $P_1, P_2, \ldots, P_n$  of distinct points in the plane is called *good* if no three of them are collinear, the polyline  $P_1P_2 \ldots P_n$  is non-self-intersecting and the triangle  $P_iP_{i + 1}P_{i + 2}$  is oriented counterclockwise for every  $i = 1, 2, \ldots, n - 2$ .
For every integer  $n \geq 3$  determine the greatest possible integer  $k$  with the following property: there exist  $n$  distinct points  $A_1, A_2, \ldots, A_n$  in the plane for which there are  $k$  distinct permutations  $\sigma : \{1, 2, \ldots, n\} \to \{1, 2, \ldots, n\}$  such that  $A_{\sigma(1)}, A_{\sigma(2)}, \ldots, A_{\sigma(n)}$  is good.
(A polyline  $P_1P_2 \ldots P_n$  consists of the segments  $P_1P_2, P_2P_3, \ldots, P_{n - 1}P_n$ .)

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