
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
Given the polynomial \(a_{0} x^{n}+a_{1} x^{n-1}+\cdots+a_{n-1} x+a_{n}\), where \(n\) is a non-negative integer, \(a_{0}\) is a positive integer, and the remaining \(a\) are integers or zero, let \(h = n + a_{0} + |a_{1}| + |a_{2}| + \cdots + |a_{n}|\). Determine the number of possible polynomials when \(h = 3\).

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