
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 and $k$ a constant such that \( k \neq 0 \).  The function $f : S \to S$ satisfies the following two properties:

(i) \( f \left( \frac{1}{x} \right) = \cos(kx) \cdot xf(x) \) 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 \( x, y \in S \) such that \( x + y \in S \).

Determine the value of $f(1)$.

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