
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]


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Given the polynomial product
\[(1 - z)^{c_1} (1 - z^2)^{c_2} (1 - z^3)^{c_3} \dotsm (1 - z^{30})^{c_{30}},\]
where the coefficients $c_k$ are positive integers. When we multiply it out and keep only terms with powers of $z$ not exceeding 30, the product is simplified to $1 - 3z$. Determine $c_{30}$.

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