
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 each positive integer  $n$ , let  $S_n = \sum_{k=1}^nk^3$ , and let  $d(n)$  be the number of positive divisors of  $n$ .  For how many positive integers  $m$ , where  $m\leq 25$ , is there a solution  $n$  to the equation  $d(S_n) = m$ ?

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