
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 $a$ is a real number, the function $f(x) = a\ln x + x^2 - 4x$.
(1) If $a = 1$, find the equation of the tangent line to the function $f(x)$ at $x = 1$.
(2) Let $g(x) = (a-2)x$. Find the range of the real number $a$ such that $f(x) \geq g(x)$ holds for all $x \in [\frac{1}{e}, e]$.
(3) Definition: If there exist two points $A$, $B$ on the graph of function $m(x)$, and let $P(x_0, y_0)$ be the midpoint of the segment $AB$. If the tangent line $l$ of $m(x)$ at point $Q(x_0, m(x_0))$ is parallel or coincident with the line $AB$, then the function $m(x)$ is called a "mean value function", and the tangent line $l$ is called the "mean value tangent" of the function $m(x)$. Determine whether the function $f(x)$ is a "mean value function". If it is, find the number of "mean value tangents" of function $f(x)"; if not, explain why.

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