
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
Let $f(x)=\log_{a}x(a>0$, and $a\neq 1)$, and its graph passes through the point $({\sqrt{e}},\frac{1}{2})$. Also, the graph of $g(x)$ is symmetric with respect to the line $y=x$ compared to the graph of $f(x)$.

$(1)$ If the range of $f(x)$ on the interval $[{\sqrt{e}},c]$ is $\left[m,n\right]$, and $n-m=\frac{3}{2}$, find the value of $c$.

$(2)$ If $g(2m)=4$ and $g(n)=25$, find the value of $2m+n$.

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