Poly polynomials

Level pending

Let \(P_1(x) = x^2 - 2\) and \(P_i(x) = P_1 (P_{i-1}(x))\) for \(i \geq 2\). Let \(N\) denote the number of real roots of \(P_{2014}(x)\). Find the first three digits of \(\log_2{N}\).

Details and assumptions

Each root is counted once, regardless of it's multiplicity.

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