
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
Determine all pairs $(f, g)$ of functions $f, g: \mathbb{R} \rightarrow \mathbb{R}$ such that for all $x, y \in \mathbb{R}$

- $f(x) \geq 0$,
- $f(x+g(y)) = f(x) + f(y) + 2 y g(x) - f(y - g(y))$.

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