Periodic fixed point

Algebra Level 5

Consider the sequence: \(x_{n+1} = 4x_n(1 − x_n)\)

Call a point \(x_0\in[0, 1]\) is \(r−\)periodic if \(x_r=x_0\). For example, \(x_0 = 0\) is always a \(r−\)periodic fixed point for any \(r\).

Let \(N\) be the number of positive \(2015−\)periodic fixed points.

Find the last 3 digits of \(N\).

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