Algebra Level 5

The sequence $\{a_n\}$ satisfies $a_0=1, a_1=213,$ and $a_n=2a_{n-1}+a_{n-2}$ for all $n \geq 2$. Let $S = \sum_{i=1}^{\infty} \frac{a_{i-1}}{a_i^2-a_{i-1}^2}.$ What is the value of $\frac{1}{S}$?

This problem is shared by Muhammad A.

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