
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
Suppose that the positive divisors of a positive integer  $n$  are  $1=d_1<d_2<\ldots<d_k=n$ , where  $k \geq 5$ . Given that  $k \leq 1000$  and  $n={d_2}^{d_3}{d_4}^{d_5}$ , compute, with proof, all possible values of  $k$ .

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