
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
find all pairs of relatively prime natural numbers  $ (m,n) $  in such a way that there exists non constant polynomial f satisfying \[ gcd(a+b+1, mf(a)+nf(b)  > 1 \]
for every natural numbers  $ a $  and  $ b $ 

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