
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
Given a positive integer $n$ such that $n \geq 2$, find all positive integers $m$ for which the following condition holds for any set of positive real numbers $a_1, a_2, \cdots, a_n$ that satisfy:
$$
a_{1} \cdot a_{2} \cdot \cdots \cdot a_{n}=1
$$

It must follow that the inequality:
$$
a_{1}^{m}+a_{2}^{m}+\cdots+a_{n}^{m} \geq \frac{1}{a_{1}}+\frac{1}{a_{2}}+\cdots+\frac{1}{a_{n}}
$$

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