
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 $f(x)=\ln x+\frac{1}{2}(a-x)^{2}$, where $a\in \mathbb{R}$.  
$(1)$ When $a=1$, find the equation of the tangent line to the function $f(x)$ at $(1,f(1))$.  
$(2)$ Discuss the monotonicity of the function $f(x)$.  
$(3)$ If $f(x)$ has two extreme points $x_{1}$ and $x_{2}$ ($x_{1} \lt x_{2}$) and the range of $|f(x_{2})-f(x_{1})|$ is $(\frac{3}{4}-\ln 2,\frac{15}{8}-2\ln 2)$, find the range of values for $a$.

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