
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 greater than -1. Find all functions \( f : S \to S \) such that \( f(x + f(y) + x f(y)) = y + f(x) + y f(x) \) for all \( x \) and \( y \), and \( \frac{f(x)}{x} \) is strictly increasing on each of the intervals \( -1 < x < 0 \) and \( 0 < x \).

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