
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 product
\[(1 - z)^{c_1} (1 - z^2)^{c_2} \dotsm (1 - z^{15})^{c_{15}},\] where $c_k$ are non-negative integers. This product has the property that when multiplied out and terms involving $z$ to a power of 20 or higher are discarded, the result is just $1 - 3z.$ Determine $c_{15}$.

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