
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
A sequence  $a_1, a_2,\dots, a_n$  of positive integers is *alagoana*, if for every  $n$  positive integer, one have these two conditions
I-  $a_{n!} = a_1\cdot a_2\cdot a_3\cdots a_n$ II- The number  $a_n$  is the  $n$ -power of a positive integer.
Find all the sequence(s) *alagoana*.

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