
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
Determine for what  $n\ge 3$  integer numbers,  it is possible to find positive integer numbers  $a_1 < a_2 < ...< a_n$  such  $\frac{1}{a_1}+\frac{1}{a_2}+...+\frac{1}{a_n}=1$  and  $a_1 a_2\cdot\cdot\cdot a_n$  is a perfect square.

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