
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
The sequence $(a_n)$ is defined recursively by $a_0=1$, $a_1=\sqrt[23]{3}$, and $a_n=a_{n-1}a_{n-2}^3$ for $n\geq 2$. 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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