
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
Consider the polynomial \((1-z)^{b_{1}} \cdot (1-z^{2})^{b_{2}} \cdot (1-z^{3})^{b_{3}} \cdots (1-z^{32})^{b_{32}}\), where \(b_{i}\) are positive integers for \(i = 1, 2, \ldots, 32\), and this polynomial has the following remarkable property: after multiplying it out and removing the terms where \(z\) has a degree higher than 32, exactly \(1 - 2z\) remains. Determine \(b_{32}\). The answer can be expressed as the difference between two powers of 2.
(1988 USA Mathematical Olympiad problem)

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