
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 all real numbers except -1 and 0. Define functions $f : S \to \mathbb{R}$ and $g: S \to \mathbb{R}$ such that:
(i) $f(2) = 1,$
(ii) $f\left(\frac{1}{x+y+1}\right) = f\left(\frac{1}{x+1}\right) + f\left(\frac{1}{y+1}\right)$ for all $x, y \in S$ where $x+y+1 \in S$,
(iii) $(x+y+1) f(x+y+1) = (x+1)(y+1) f(x+1)f(y+1)$ for all $x, y \in S$ where $x+y+1 \in S$,
(iv) $g(x) = f(x+1)$.
Find the number of possible functions $f(x)$ and $g(x).$

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