
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 functions $f(x)=\ln x$ and $g(x)=a(x-1)^2-1$.  
$(1)$ When $a=\frac{1}{4}$, find the maximum value of the function $F(x)=f(x)-g(x)$.  
$(2)$ When $a=-\frac{1}{4}$, find the equation of the common tangent line to the curves $y=f(x)$ and $y=g(x)$.

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