
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 an integer $n \geq 3$, define the sequence $\alpha_{1}, \alpha_{2}, \ldots, \alpha_{k}$ by the prime factorization of $n!$ as $n! = p_{1}^{\alpha_{1}} p_{2}^{\alpha_{2}} \ldots p_{k}^{\alpha_{k}}$, where the primes $p_{1}<p_{2}<\ldots <p_{k}$ correspond to the respective exponents $\alpha_{1}, \alpha_{2}, \ldots, \alpha_{k}$. Determine all integers $n \geq 3$ for which $\alpha_{1}, \alpha_{2}, \ldots, \alpha_{k}$ form a geometric sequence.

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