
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 $a=1$, find the monotonic intervals of the function $f(x)$.
(2) When $a > 0$, if the minimum value of $f(x)$ in the interval $[1,e]$ is $-2$, find the range of $a$.
(3) If for any $x_{1}$, $x_{2}∈(0,+∞)$, $x_{1} < x_{2}$ and $f(x_{1})+2x_{1} < f(x_{2})+2x_{2}$ always holds, find the range of $a$.

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