
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 polynomials  $P(x)$  with real coefficents, such that for all  $x,y,z$  satisfying  $x+y+z=0$ , the equation below is true: \[P(x+y)^3+P(y+z)^3+P(z+x)^3=3P((x+y)(y+z)(z+x))\]

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