Consider a sequence $x_1, x_2, \cdots, x_{12}$ of real numbers such that $x_1 = 1$ and for $n = 1, 2, \dots, 10$, let \[ x_{n+2} = \frac{(x_{n+1} + 1)(x_{n+1} - 1)}{x_n}. \] Suppose $x_n > 0$ for $n = 1, 2, \dots, 11$ and $x_{12} = 0$. Then the value of $x_2$ can be written as $\frac{\sqrt{a} + \sqrt{b}}{c}$ for positive integers $a, b, c$ with $a > b$ and no square dividing $a$ or $b$. Find $100a + 10b + c$.

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