
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
We are given a natural number  $d$ . Find all open intervals of maximum length  $I \subseteq R$  such that for all real numbers  $a_0,a_1,...,a_{2d-1}$  inside interval  $I$ , we have that the polynomial  $P(x)=x^{2d}+a_{2d-1}x^{2d-1}+...+a_1x+a_0$  has no real roots.

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