
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
Given a polynomial of the form 
\[(1 - z)^{b_1} (1 - z^2)^{b_2} (1 - z^3)^{b_3} \dotsm (1 - z^{20})^{b_{20}},\]
where the \(b_k\) are positive integers, and the polynomial simplifies such that when all terms involving \(z\) to a power higher than 20 are discarded, the result is \(1 - 3z\). Determine \(b_{20}\).

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