
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
Let $S$ be the set of all nonzero real numbers. The function $f : S \to S$ satisfies the following two properties:

(i) 
\[ f \left( \frac{1}{x} \right) = x^2 f(x) \]
for all $x \in S.$

(ii) 
\[ f \left( \frac{1}{x} \right) + f \left( \frac{1}{y} \right) = 2 + f \left( \frac{1}{x + y} \right) \]
for all $x \in S$ and $y in S$ such that $x + y \in S.$

Determine the number $n$ of possible values of $f(1)$ and the sum $s$ of all possible values of $f(1)$. Compute $n \times s.$

Present the answer in LaTex format: \boxed{Your answer}
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