
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  $k \ge 5$  for which there is a positive integer  $n$  with exactly  $k$  positive divisors $1 = d_1 <d_2 < ... <d_k = n$  and  $d_2d_3 + d_3d_5 + d_5d_2 = n$ .

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