
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
We define \(\mathbb{F}_{101}[x]\) as the set of all polynomials in \(x\) with coefficients in \(\mathbb{F}_{101}\) (the integers modulo 101 with usual addition and subtraction), so that two polynomials are equal if and only if the coefficients of \(x^{k}\) are equal in \(\mathbb{F}_{101}\) for each nonnegative integer \(k\). For example, \((x+3)(100 x+5)=100 x^{2}+2 x+15\) in \(\mathbb{F}_{101}[x]\) because the corresponding coefficients are equal modulo 101.

We say that \(f(x) \in \mathbb{F}_{101}[x]\) is lucky if it has degree at most 1000 and there exist \(g(x), h(x) \in \mathbb{F}_{101}[x]\) such that
$$
f(x)=g(x)\left(x^{1001}-1\right)+h(x)^{101}-h(x)
$$
in \(\mathbb{F}_{101}[x]\). Find the number of lucky polynomials.

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