
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  $r_1$ ,  $r_2$ ,  $\ldots$ ,  $r_{20}$  be the roots of the polynomial  $x^{20}-7x^3+1$ .  If \[\dfrac{1}{r_1^2+1}+\dfrac{1}{r_2^2+1}+\cdots+\dfrac{1}{r_{20}^2+1}\] can be written in the form  $\tfrac mn$  where  $m$  and  $n$  are positive coprime integers, find  $m+n$ .

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