
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
Define the sequence $(a_n)$ recursively as follows: $a_0 = 1$, $a_1 = \sqrt[17]{3}$, and $a_n = a_{n-1} a_{n-2}^2$ for $n \geq 2$. Find the smallest positive integer $k$ such that $a_1a_2 \cdots a_k$ is an integer.

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