
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 
$$
f(x) = a \ln x + \frac{1}{2} x^2 + (a + 1)x + 1.
$$

1. When $a = -1$, find the interval where the function $f(x)$ is monotonically increasing.
2. If the function $f(x)$ is an increasing function on the interval $(0, +\infty)$, find the range of values for the real number $a$.
3. If $a > 0$, and for any $x_1, x_2 \in (0, +\infty)$ with $x_1 \neq x_2$, it holds that $|f(x_1) - f(x_2)| > 2|x_1 - x_2|$, find the minimum value of the real number $a$.

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