
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
The sequence  $f_0, f_1, \dots$  of polynomials in  $\mathbb{F}_{11}[x]$  is defined by  $f_0(x) = x$  and  $f_{n+1}(x) = f_n(x)^{11} - f_n(x)$  for all  $n \ge 0$ . Compute the remainder when the number of nonconstant monic irreducible divisors of  $f_{1000}(x)$  is divided by  $1000$ .

*Proposed by Ankan Bhattacharya*

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