
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 \( f(x) = \frac{2x - a}{x^2 + 2} \) ( \( x \in \mathbb{R} \) ) is an increasing function on the interval \([-1, 1]\):

(I) Find the set of values for the real number \( a \), denoted as set \( A \).

(II) Suppose the equation \( f(x) = \frac{1}{x} \) has two roots \( x_1 \) and \( x_2 \). Determine whether there exists a real number \( m \) such that the inequality \( m^2 + t m + 1 \geq \left| x_1 - x_2 \right| \) holds for any \( a \in A \) and \( t \in [-1, 1] \). If such an \( m \) exists, find the range of values \( m \) can take. If not, provide a reasoning.

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