
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 $f(x)= \frac {1}{2}x^{2}$, $g(x)=e\ln x$.  
(I) Let the function $F(x)=f(x)-g(x)$, find the intervals of monotonicity for $F(x)$;  
(II) If there exist constants $k$, $m$, such that $f(x)\geqslant kx+m$ always holds for $x\in\mathbb{R}$, and $g(x)\leqslant kx+m$ always holds for $x\in(0,+\infty)$, then the line $y=kx+m$ is called the "boundary line" between the functions $f(x)$ and $g(x)$. The question is: does a "boundary line" exist between $f(x)$ and $g(x)$? If it exists, find the equation of the "boundary line"; if not, explain why.

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