
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
Let $a_1, a_2, \ldots$ be a sequence defined by:
(i) $a_1 = 1$, 
(ii) $a_{3n} = n^2 \cdot a_n$ for any positive integer $n$.
Determine the value of $a_{3^{100}}$.

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