
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 the polynomial
$$
(1-z)^{b_{1}}(1-z^{2})^{b_{2}}(1-z^{3})^{b_{3}} \cdots(1-z^{32})^{b_{32}}
$$
where \( b_{i} \) are positive integers for \( i = 1, 2, \ldots, 32 \), and this polynomial has a remarkable property: after expanding it and removing the terms with \( z \) raised to powers higher than 32, what remains is \( 1 - 2z \). Determine \( b_{32} \) (the answer can be expressed as the difference of two powers of 2).

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