A function $f$ has its domain equal to the set of integers $0, 1, \ldots, 11$, and $f(n) \geq 0$ for all such $n$. The function $f$ satisfies the following conditions:
- $f(0) = 0$
- $f(6) = 1$
- If $x \geq 0$, $y \geq 0$, and $x + y \leq 11$, then $f(x + y) = \frac{f(x) + f(y)}{1 - f(x)f(y)}$

Find $f(2)^2 + f(10)^2$.

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