jmc

Let $S$ be the set of all nonzero real numbers. The function $f:S\to S$ 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 $n\times s.$