
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
In the polynomial \((1-z)^{b_{1}} \cdot (1-z^{2})^{b_{2}} \cdot (1-z^{3})^{b_{3}} \cdots \cdot (1-z^{32})^{b_{32}}\), where \(b_{i}\) are positive integers (\(i=1,2,\cdots,32\)), this polynomial has the peculiar property that after expansion, if the terms with degrees higher than 32 in \(z\) are removed, the result is precisely \(1 - 2z\). 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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