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

(ii) \(f \left( \frac{1}{x} \right) +  f \left( \frac{1}{y} \right) = 1 + f \left( \frac{1}{x + y} \right)\) for all \(x, y \in S\) and \(x + y \in S.\)

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

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