Let $x_1$, $x_2$, and $x_3$ be the roots of the polynomial $x^3+3x+1$.  There are relatively prime positive integers $m$ and $n$ such that $\tfrac{m}{n}=\tfrac{x_1^2}{(5x_2+1)(5x_3+1)}+\tfrac{x_2^2}{(5x_1+1)(5x_3+1)}+\tfrac{x_3^2}{(5x_1+1)(5x_2+1)}$.  Find $m+n$.

Let's think step by step and output the final answer within \boxed{}.
user