
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 $f(x) = ax^2 - (a+2)x + \ln x$,
1. When $a=1$, determine the intervals of monotonicity for the function $f(x)$.
2. When $a > 0$, if the minimum value of $f(x)$ on the interval $[1, e]$ is $-2$, find the range of possible values for $a$.
3. If for any $x_1, x_2 \in (0, +\infty)$ with $x_1 < x_2$ it holds that $f(x_1) + 2x_1 < f(x_2) + 2x_2$, find the range of possible values for $a$.

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