
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 $f(x)=e^{x}-ax$ and $g(x)=\ln x-ax$, where $a\in \mathbb{R}$.  
$(1)$ When $a=1$, find the monotonic interval of the function $y=f(g(x))$.  
$(2)$ Let $h(x)=f(x)-g(x)$ and the minimum value of $h(x)$ be $m$. Find the minimum value of the function $F(x)=e^{x}-e^{m}\ln x$.  
(Where $e\approx 2.71828$ is the base of the natural logarithm)

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