
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
**Problem 6.** Let  $m\geq 5$  and  $n$  are given natural numbers, and  $M$  is regular  $2n+1$ -gon. Find the number of the convex  $m$ -gons with vertices among the vertices of  $M$ , who have at least one acute angle.

*Alexandar Ivanov*

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