
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
Determine all pairs of non-constant polynomials  $p(x)$  and  $q(x)$ , each with leading coefficient  $1$ , degree  $n$ , and  $n$  roots which are non-negative integers, that satisfy  $p(x)-q(x)=1$ .

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