
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
Let  $n$  be a positive integer, and define  $f(n)=1!+2!+\ldots+n!$ . Find polynomials  $P$  and  $Q$  such that  $$ f(n+2)=P(n)f(n+1)+Q(n)f(n) $$  for all  $n\ge1$ .

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