
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
Let $a$, $b\in R$, and functions $f\left(x\right)=\ln x-ax$ and $g(x)=\frac{b}{x}$.<br/>$(Ⅰ)$ If $f\left(x\right)=\ln x-ax$ and $g(x)=\frac{b}{x}$ have a common point $P\left(1,m\right)$, and the tangents at point $P$ are the same, find the equation of the tangent line at point $P$;<br/>$(Ⅱ)$ If the function $f\left(x\right)$ has extremum but no zeros, find the range of real number $a$;<br/>$(Ⅲ)$ When $a \gt 0$, $b=1$, find the minimum value of $F\left(x\right)=f\left(x\right)-g\left(x\right)$ on the interval $\left[1,2\right]$.

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