
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
Given integers \( m \) and \( n \) where \( 4 < m < n \), and \( A_{1} A_{2} \cdots A_{2n+1} \) is a regular \( 2n+1 \)-gon with vertices set \( P = \{A_{1}, A_{2}, \cdots, A_{2n+1}\} \). Determine the number of convex \( m \)-gons with vertices in \( P \) that have exactly two acute internal angles.

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