
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)
\[ f\left(\frac{1}{x}\right) = xf(x) \]
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 \( x + y \in S \).

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 \( n \times s \).

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