
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
If a positive function \( f(x) \) defined on a set \( D \) satisfies the following inequality for any \( x, y \in D \):
$$
f(x) f(y) \leqslant\left[f\left(\frac{x+y}{2}\right)\right]^{2}
$$
(with equality if and only if \( x = y \)), then for any \( x_1, x_2, \ldots, x_n \in D \), the following inequality holds:
$$
f(x_1) f(x_2) \cdots f(x_n) \leqslant\left[f\left(\frac{x_1 + x_2 + \cdots + x_n}{n}\right)\right]^n
$$
(with equality if and only if \( x_1 = x_2 = \cdots = x_n \)).

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