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