
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,  $(x+3)(100x+5)=100x^2+2x+15$  in  $\mathbb F_{101}[x]$  because the corresponding coefficients are equal modulo  $101$ .

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

*Proposed by Michael Ren.*

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