Let \(f(x)\) be a strictly increasing function defined for all \(x > 0\) such that \(f(x) > -\frac{1}{x}\) for all \(x > 0\), and \(f(x) f \left( f(x) + \frac{1}{x} \right) = 1\) for all \(x > 0\). Find \(f(1)\).
