
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 $g : S \to \mathbb{R}$ be a function such that:
(i) $g(1) = 2,$
(ii) $g\left(\frac{1}{x+y}\right) = g\left(\frac{1}{x}\right) + g\left(\frac{1}{y}\right)$ for all $x, y \in S$ such that $x + y \in S,$ and
(iii) $(x+y) g(x+y) = x^2 y^2 g(x) g(y)$ for all $x, y \in S$ such that $x + y \in S.$

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

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