
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 positive real numbers. Define $f : S \to \mathbb{R}$ such that
(i) $f(1) = 0,$
(ii) $f \left( \frac{1}{x} + \frac{1}{y} \right) = f(x) + f(y)$ for all $x, y \in S,$
(iii) $\log(x + y) f(x+y) = \log(x)f(x) + \log(y)f(y)$ for all $x, y \in S.$ 

Find the number of possible functions $f(x).$

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