
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
In a triangle $C$, the inequality $\frac{1}{A} + \frac{1}{B} + \frac{1}{C} \geq \frac{9}{\pi}$ holds. In a quadrilateral $ABCD$, the inequality $\frac{1}{A} + \frac{1}{B} + \frac{1}{C} + \frac{1}{D} \geq \frac{16}{2\pi}$ holds. In a convex pentagon $ABCDE$, the inequality $\frac{1}{A} + \frac{1}{B} + \frac{1}{C} + \frac{1}{D} + \frac{1}{E} \geq \frac{25}{3\pi}$ holds. Based on the above, guess what is the general form of the inequality that holds for an $n$-sided polygon $A_1A_2\ldots A_n (n \geq 3)$.

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