
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 $f(x)=-x^{2}+2ex+t-1$, $g(x)=x+ \frac{{e}^{2}}{x} (x > 0)$, where $e$ represents the base of the natural logarithm.

$(1)$ If $g(x)=m$ has a root, find the range of $m$;

$(2)$ Determine the range of $t$ such that $g(x)-f(x)=0$ has two distinct real roots.

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