
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 $f(x)=ax^{2}-\ln x+(2a-1)x$, where $a\in \mathbb{R}$.  
$(1)$ Discuss the monotonicity of $f(x)$.  
$(2)$ Let $a \gt 0$. If the inequality $f(x)+\frac{e}{2}≥0$ holds for $x\in (0,+\infty)$, find the range of values for $a$.

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