Do You See The Connection?

Algebra Level 5

Let the sequence of real numbers \( {x_n} \) be defined with

\[\begin{cases} x_1=k\\ x_{n+1} = 4x_n(1-x_n),\ \ n \geq 1. \end{cases}\]

Let there be \( N \) distinct values of \( k \) such that \( x_{2014} = 0 \). Find the last three digits of \( N \).

Note: For those interested, this question was partly inspired by the function \( f(x) = \lambda x(1-x) \) also know as the logistic function and is related to population growth.

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