Solve the following math problem efficiently and clearly:

- For simple problems (2 steps or fewer):
Provide a concise solution with minimal explanation.

- For complex problems (3 steps or more):
Use this step-by-step format:

## Step 1: [Concise description]
[Brief explanation and calculations]

## Step 2: [Concise description]
[Brief explanation and calculations]

...

Regardless of the approach, always conclude with:

Therefore, the final answer is: $oxed{answer}$. I hope it is correct.

Where [answer] is just the final number or expression that solves the problem.

Problem: 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 $n \times s.$
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