
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 $f(x)=a\ln x$, where $a \gt 0$.  
$(1)$ If the line $y=\frac{x}{{e}^{a}}+{e}^{2}$ is a tangent line to the curve $y=f(x)$, find the value of the real number $a$.  
$(2)$ If for any two distinct positive real numbers $m$ and $n$, $\sqrt{mn}+\frac{m+n}{2}>\frac{m-n}{f(m)-f(n)}$, find the range of values for the real number $a$.

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