
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 $g(x)=e^{x}$ ($e^{x}=2.718\ldots$, where $e$ is the base of natural logarithm).  
$(1)$ Find the monotonic intervals of the function $F(x)=f(x)-g(x-1)$.  
$(2)$ If the inequality $xf(x)-k(x+1)f[g(x-1)]\leqslant 0$ holds for all $x\in[1,+\infty)$, find the range of real number $k$.

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