Suppose a function \(f: \mathbb{R} \to \mathbb{R}\) has the satisfies \(f(x+a) = \frac{1}{2} + \sqrt{ f(x) - (f(x))^2 } \) for all \(x.\)

If the period of \(f(x)\) is \( n \times a \), then find \(n.\)

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