
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 a sequence \((a_n)\) recursively by setting \(a_0 = 1\), \(a_1 = \sqrt[17]{3}\), and for \(n \geq 2\), \(a_n = a_{n-1}^3 a_{n-2}^2\). Determine the smallest positive integer \(k\) such that the product \(a_1a_2\cdots a_k\) is an integer.

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