
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 the function f(x) = $$\begin{cases} \frac{a}{x} + 2, x > 1 \\ -x^2 + 2x, x \leq 1 \end{cases}$$, when x₁ ≠ x₂, $$\frac{f(x_1) - f(x_2)}{x_1 - x_2} > 0$$, determine the range of value for 'a'.

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