
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 integers \(n \geq 3\) such that the following property holds: if we list the divisors of \(n!\) in increasing order as \(1 = d_1 < d_2 < \cdots < d_k = n!\), then
\[ d_2 - d_1 \leq d_3 - d_2 \leq \cdots \leq d_k - d_{k-1}. \]

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