
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 the largest  $a$  for which there exists a polynomial  $$ P(x) =a x^4 +bx^3 +cx^2 +dx +e $$  with real coefficients which satisfies  $0\leq P(x) \leq 1$  for  $-1 \leq x \leq 1.$ 

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