
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 $f(x)=\frac{1}{2}x^2-(a+2)x+2a\ln x (a \in \mathbb{R})$.  
$(1)$ If $a \gt 0$, discuss the monotonicity of the function $f(x)$.  
$(2)$ Let $g(x)=- (a+2)x$. If there exists at least one $x_{0} \in [e,4]$ such that $f(x_{0}) > g(x_{0})$, find the range of real numbers for $a$.

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