
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 $f(x) = \ln x - \frac{1}{2}ax^2 + ax$, where $a \in \mathbb{R}$.

$(1)$ When $a < 0$, discuss the number of extreme points of the function $f(x)$.

$(2)$ If the inequality $f(x) \leqslant 2ax - x - 1$ always holds for $x$, find the minimum value of the integer $a$.

$(3)$ For any two given points $A(x_1, f(x_1))$ and $B(x_2, f(x_2))$ on the graph of the function $f(x)$, try to determine the relationship between $f\left( \frac{x_1 + x_2}{2}\right)$ and $\frac{f(x_1) - f(x_2)}{x_2 - x_1}$ (where $f′(x)$ is the derivative of the function $f(x)$), and provide a proof.

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