
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
Find all functions \( f \) from the set \(\mathbf{R}\) of real numbers into \(\mathbf{R}\) which satisfy for all \( x, y, z \in \mathbf{R} \) the identity

\[ 
f(f(x)+f(y)+f(z)) = f(f(x)-f(y)) + f(2xy + f(z)) + 2f(xz-yz). 
\]

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