
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  $n$  be given,  $n\ge 4,$  and suppose that  $P_1,P_2,\dots,P_n$  are  $n$  randomly, independently and uniformly, chosen points on a circle. Consider the convex  $n$ -gon whose vertices are the  $P_i.$  What is the probability that at least one of the vertex angles of this polygon is acute.?

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