
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 functions  $f: \mathbb{R} \times \mathbb{R} \to \mathbb{R}$  such that
\[ f(x,y) + f(y,z) + f(z,x) = \max \{ x,y,z \} - \min \{ x,y,z \} \] for every  $x,y,z \in \mathbb{R}$ and there exists some real  $a$  such that  $f(x,a) = f(a,x) $  for every  $x \in \mathbb{R}$ .

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