
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
Define $S$ as the set of all nonzero real numbers. Suppose $f : S \to S$ satisfies:
(i) \(f\left(\frac{1}{x}\right) = \frac{1}{k}xf(x)\) for all \(x \in S.\)
(ii) \(f\left(\frac{1}{x}\right) + f\left(\frac{1}{y}\right) = k + f\left(\frac{1}{kx + ky}\right)\) for all \(x, y \in S\) where \(x + y \in S.\)

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

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