
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 \( n \geq 5 \) be an integer. Find the largest integer \( k \) (as a function of \( n \)) such that there exists a convex n-gon \( A_1 A_2 \ldots A_n \) for which exactly \( k \) of the quadrilaterals \( A_i A_{i+1} A_{i+2} A_{i+3} \) have an inscribed circle, where indices are taken modulo \( n \).

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