
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
Find all positive integers $n$ such that for any $a_1, a_2, \ldots, a_n$ where $a_i \in (0, 1]$ for $1 \leq i \leq n$, the inequality $\sum_{i=1}^{n} \frac{\sqrt{1-a_{i}}}{a_{i}} \leq \frac{1}{\prod_{i=1}^{n} a_{i}}$ always holds.

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