+46 Let be S the set of all nonzero real numbers. The function \(f : S \to \mathbb{R}\) satisfies the following two properties:
(i) First,
\(f\left( \frac{1}{x} \right) = xf(x)\) for all \(x \in S.\)
(ii) Second,
\(f \left( \frac{1}{x} \right) + f \left( \frac{1}{y} \right) = 1 + f \left( \frac{1}{x + y} \right)\) for all \(x \in S\) and \(y \in S\) such that \(x + y \in S.\)
Let n be the number of possible values of \(f(1),\) and let s be the sum of all possible values of \(f(1).\) Find \(n \times s.\)
