
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
For integers $n \geq 3$, let $n! = \prod_{i=1}^{k} p_{i}^{\alpha_{i}}$, where $p_{1}, p_{2}, \cdots, p_{k}$ are prime numbers, and $p_{1}<p_{2}<\cdots<p_{k}$, and $\alpha_{1}, \alpha_{2}, \cdots, \alpha_{k}$ are positive integers. Find all $n \geq 3$ such that $\alpha_{1}, \alpha_{2}, \cdots, \alpha_{k}$ form a geometric progression.

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