
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  $a, b$  be given two real number with  $a \ne 0$ . Find all polynomials  $P$  with real coefficients such that  $x P(x - a) = (x - b)P(x)$  for all  $x\in  R$ 

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