
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 $R$ be the set of all real numbers. Find all functions $f: \mathbf{R} \rightarrow \mathbf{R}$ that satisfy the following equation for any real numbers $x$ and $y$:
$$
f(x+f(x+y))+f(xy) = x+f(x+y)+y f(x).
$$

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