
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 $f(x)=x-\ln x$.
$(1)$ If the tangent line of the curve $y=f(x)$ at $x=x_0$ passes through the origin, find $x_0$ and the equation of this tangent line;
$(2)$ Let $g(x)=(e-1)x$, if the function $F(x)= \begin{cases} f(x), & x\geqslant a \\ g(x), & x < a \end{cases}$ has the range $\mathbb{R}$, find the range of the real number $a$.

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