Let \(c\) be a real number. The sequence \(a_1, a_2, a_3, ... \) is defined by \(a_1 = c\) and \(a_n= 2 a_{n-1}^2 -1\) for all \(n \geq 2\). Find the sum of \(c\) such that \(a_n < 0\) for all \(n \geq 1.\)

This problem is not original.

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