
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
Define polynomials  $f_{n}(x)$  for  $n \geq 0$  by  $f_{0}(x)=1, f_{n}(0)=0$  for  $n \geq 1,$  and  $$ \frac{d}{d x} f_{n+1}(x)=(n+1) f_{n}(x+1) $$  for  $n \geq 0 .$  Find, with proof, the explicit factorization of  $f_{100}(1)$  into powers of distinct primes.

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