
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
Given a polynomial \( P(x) = a_{2n} x^{2n} + a_{2n-1} x^{2n-1} + \ldots + a_1 x + a_0 \) where each coefficient \( a_i \) belongs to the interval \([100,101]\), what is the smallest natural number \( n \) such that this polynomial can have a real root?

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