
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 coefficients such that if real numbers  $x,y$  and  $z$  satisfy  $x+y+z=0,$  then the points  $\left(x,P(x)\right), \left(y,P(y)\right), \left(z,P(z)\right)$  are all colinear.

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