
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 \( r_{1}, r_{2}, \cdots, r_{20} \) be the roots of the polynomial \( x^{20}-7x^{3}+1 \). If \(\frac{1}{r_{1}^{2}+1}+\frac{1}{r_{2}^{2}+1}+\cdots+\frac{1}{r_{20}^{2}+1} \) can be expressed in the form \( \frac{m}{n} \) (with \( m \) and \( n \) coprime), find the value of \( m+n \).

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