
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 positive real numbers excluding one, and let $g : T \to \mathbb{R}$ be a function such that:

(i) $g(e) = 1$,
(ii) $g\left(\frac{1}{e^{x+y}}\right) = g\left(\frac{1}{e^x}\right) + g\left(\frac{1}{e^y}\right)$ for all $x, y \in \mathbb{R}$,
(iii) $(e^{x+y} - 1) g(e^{x+y}) = (e^x - 1)(e^y - 1) g(e^x) g(e^y)$ for all $x, y \in \mathbb{R}$.

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

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