
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 $f\left(x\right)=\frac{{\ln x}}{{1-x}}$, $x\in D$, where $D=\left(0,1\right)\cup \left(1,+\infty \right)$. <br/>$(1)$ Find the equation of the tangent line to the function $f\left(x\right)$ at the point $({\frac{1}{2}},f({\frac{1}{2}}))$; <br/>$(2)$ If $g\left(x\right)=-\frac{a}{{\sqrt{x}}}$, and $\forall x\in D$, $f\left(x\right)\geqslant g\left(x\right)$ always holds, find the range of values for $a$.

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