
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) = x² + a*ln(x) + 1, if ∀x₁, x₂ ∈ [3, +∞), (x₁ ≠ x₂), ∃a ∈ [1, 2], such that $\frac{f(x₁) - f(x₂)}{x₂ - x₁} < m$, determine the minimum value of the real number m.

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