
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 that $a$ is a real number, the function $f(x)=a\ln x+x^{2}-4x$.

$(1)$ When $a=1$, find the equation of the tangent line to the function $f(x)$ at $x=1$;

$(2)$ Let $g(x)=(a-2)x$, if for all $x\in\left[\frac{1}{e},e\right]$, $f(x)\geqslant g(x)$ holds, find the range of the real number $a$;

$(3)$ Definition: If there exist two points $A$, $B$ on the graph of function $m(x)$, and let the midpoint of segment $AB$ be $P(x_{0},y_{0})$, if the tangent line $l$ at point $Q(x_{0},m(x_{0}))$ is parallel or coincident with line $AB$, then the function $m(x)$ is called a "mean value average function", and the tangent line $l$ is called the "mean value average tangent line" of the function $m(x)$. Determine whether the function $f(x)$ is a "mean value average function". If it is, find the condition for the function $f(x)$'s "mean value average tangent line"; if not, explain why.

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