
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 functions $f(x)=x\ln x$ and $g(x)=\frac{x}{e^{x}}$.
(I) Let $F(x)=f(x)-g(x)$. Determine the number of zeros of $F(x)$ in the interval $(1,2)$ and explain the reasoning.
(II) Let $x\_0$ be the zero of $F(x)$ in $(1,2)$ from part (I), and define $m(x)=\min\{f(x),g(x)\}$. If $m(x)=n$ ($n\in\mathbb{R}$) has two distinct real roots $x\_1$ and $x\_2$ ($x\_1 < x\_2$) in $(1,+\infty)$, compare the sizes of $x\_1+x\_2$ and $2x\_0$, and provide a proof.

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