
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) = -$\frac{a}{2}$x² + (a+1)x - lnx (where a ∈ ℝ),
1. Find the extreme value of the function f(x) when a = 0.
2. Discuss the monotonicity of the function f(x) when a > 0.
3. If for any a ∈ (2, 3) and any x₁, x₂ ∈ [1, 2], the inequality $\frac{a^2 - 1}{2}$m + ln2 > |f(x₁) - f(x₂)| always holds, find the range of the real number m.

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