
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 nonzero real numbers. Let $f : S \to \mathbb{R}$ be a function satisfying:
  - (i) $f(1) = 3.$
  - (ii) $f \left( \frac{1}{x+y} \right) = 3 f \left( \frac{1}{x} \right) + 3 f \left( \frac{1}{y} \right)$ for all $x,$ $y \in S$ such that $x+y \in S.$
  - (iii) $(x+y+1) f(x+y) = xy f(x) f(y)$ for all $x,$ $y \in S$ such that $x+y \in S.$

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

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