
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 reals  $\lambda$  for which there is a nonzero polynomial  $P$  with real coefficients such that  $$ \frac{P(1)+P(3)+P(5)+\ldots+P(2n-1)}n=\lambda P(n) $$  for all positive integers  $n$ , and find all such polynomials for  $\lambda=2$ .

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