
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 $T$ be the set of nonzero real numbers. Let $g : T \to \mathbb{R}$ be a function such that:
(i) $g(2) = 1$,
(ii) $g \left( \frac{2}{x + y} \right) = g \left( \frac{2}{x} \right) + g \left( \frac{2}{y} \right)$ for all $x, y \in T$ such that $x + y \in T$,
(iii) $(x + y) g(x + y) = 2xy g(x) g(y)$ for all $x, y \in T$ such that $x + y \in T$.

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

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