
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 positive integers  $p,q$  with  $\gcd(p,q)=1$  such as  $p+q^2=(n^2+1)p^2+q$ . If the parameter  $n$  is a positive integer, find all possible couples  $(p,q)$ .

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