Functions help

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 $n \times s.$