
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
Let \( p(x) \) be the polynomial \( (1 - x)^a (1 - x^2)^b (1 - x^3)^c \dots (1 - x^{32})^k \), where \( a, b, \ldots, k \) are integers. When expanded in powers of \( x \), the coefficient of \( x \) is \(-2\) and the coefficients of \( x^2, x^3, \ldots, x^{32} \) are all zero. Find \( k \).

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