
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 of positive integers \( a_1, a_2, \cdots \) is called "good" if and only if for each positive integer \( n \), the following two conditions are satisfied:
1. \( a_{n!} = a_1 a_2 \cdots a_n \), e.g., 
\[ a_{24} = a_{4!} = a_1 a_2 a_3 a_4 \]
2. \( a_n \) is the \( n \)-th power of a positive integer.

Find all such "good" sequences.

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