
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  $n \geq 3$  we define the sequence  $\alpha_1, \alpha_2, \ldots, \alpha_k$  as the sequence of exponents in the prime factorization of  $n! = p_1^{\alpha_1}p_2^{\alpha_2} \ldots p_k^{\alpha_k}$ , where  $p_1 < p_2 < \ldots < p_k$  are primes. Determine all integers  $n \geq 3$  for which  $\alpha_1, \alpha_2, \ldots, \alpha_k$  is a geometric progression.

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