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Let \(f(x) = \cos\,2x\) and \(g(x) = 2x^2 - 1\).

1. Find \(f\left(\frac{\pi}{2}\right)\).
2. Find \((g \circ f)\left(\frac{\pi}{2}\right)\).
3. Given that \((g \circ f)\) can be written in the form \(\cos(kx)\), find the value of \(k\), \(k \in \mathbb{Z}\).