
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 \( P_{n}(x) = x^{n} + \ldots \) be a polynomial of degree \( n \) with a leading coefficient of 1, and suppose \( \left| P_{n}(x) \right| \leq \frac{1}{2^{n-1}} \) for \( |x| \leq 1 \). Then \( P_{n}(x) = \frac{1}{2^{n-1}} T_{n}(x) \). (In other words, the polynomial \( \frac{1}{2^{n-1}} T_{n}(x) \) is the least deviating from zero on the interval \([-1, 1]\) polynomial of degree \( n \) with a leading coefficient of 1.)

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