
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
A polynomial product of the form
\[
(1 - z)^{b_1} (1 - z^2)^{b_2} (1 - z^3)^{b_3} \dotsm (1 - z^{16})^{b_{16}},
\]
where the $b_k$ are positive integers, has the property that if we multiply it out and discard all terms involving $z$ to a power larger than 16, what is left is just $1 - 3z.$ Determine $b_{16}$.

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