
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 + ax - alnx, where a ∈ R:
1. If a = 1, find the monotonic intervals and extreme values of f(x).
2. Let g(x) = f(x) + (a + 2)lnx - (a + 2b - 2)x, and g(x) has two extreme points x₁ and x₂ (x₁ < x₂). If $$b ≥ 1 + \frac{4\sqrt{3}}{3}$$, find the minimum value of g(x₁) - g(x₂).

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