
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 $f(x)=e^{x}\ (e=2.71828…)$, where $g(x)$ is its inverse function.
(1) Let point $P(-1,0)$. Find the equation of the line that passes through point $P$ and is tangent to the curve $y=f(x)$;
(2) Find the monotonic intervals and the extreme values of the function $F(x)=\frac{x}{g(x)}$, and compare the sizes of $\sqrt{2}\ln\sqrt{3}$ and $\sqrt{3}\ln\sqrt{2}$.

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