
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 $$f(x)= \frac {2x-a}{x^{2}+2} \quad (x \in \mathbb{R})$$ is an increasing function on the interval \([-1, 1]\)  
(I) Find the range of values for the real number $a$;  
(II) Let the range of values for the real number $a$ be set $A$, and suppose the equation $$f(x)= \frac {1}{x}$$ has two non-zero real roots denoted as $x_1$, $x_2$.  
① Find the maximum value of $|x_1 - x_2|$;  
② Investigate whether there exists a real number $m$ such that the inequality $m^2 + tm + 1 > |x_1 - x_2|$ holds for all $a \in A$ and $t \in [-1, 1]$. If it exists, find the range of values for $m$; if not, explain why.

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