Let $x_1=1/20$, $x_2=1/13$, and \[x_{n+2}=\dfrac{2x_nx_{n+1}(x_n+x_{n+1})}{x_n^2+x_{n+1}^2}\] for all integers $n\geq 1$.  Evaluate $\textstyle\sum_{n=1}^\infty(1/(x_n+x_{n+1}))$.

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