
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 $(b_n)$ by $b_0=2$, $b_1=\sqrt[11]{3}$, and $b_n = b_{n-1}^{3} b_{n-2}^{2}$ for $n \geq 2$. Determine the smallest positive integer $j$ such that the product $b_1b_2\cdots b_j$ is an integer.

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