
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) = alnx - x + 1.
(1) Determine the range of values for the real number a if f(x) < 0 holds true for any x ∈ (1, +∞).
(2) When 0 < a ≤ e + $\frac{1}{e}$, if the function g(x) = f(x) + $\frac{1}{x}$ - 1 has two extreme points x₁, x₂ (x₁ < x₂), find the maximum value of g(x₂) - g(x₁).

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