
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 $f(x)=\frac{1}{2}x^{2}-4ax+a\ln x+a+\frac{1}{2}$, where $a\in \mathbb{R}$.

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

$(2)$ Let the derivative of the function $f(x)$ be denoted as $f'(x)$. If the inequality $f(x) < xf'(x)$ holds for any real number $x\in(1,+\infty)$, find the range of the real number $a$;

$(3)$ Let the function $g(x)=f(x)+2a$, and $g'(x)$ be the derivative of the function $g(x)$. If the function $g(x)$ has two extreme points $x_1$ and $x_2$, and $g(x_1)+g(x_2)\geqslant g'(x_1x_2)$, find the range of the real number $a$.

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