
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 $f(x)=\ln x$ and $g(x)= \frac {1}{2}x|x|$:  
(Ⅰ) Define $F(x)=x \cdot f(x)-g(x)$, find the intervals where $F(x)$ is monotonic;  
(Ⅱ) Suppose for any $x_1, x_2 \in [1,+\infty)$ with $x_1 < x_2$, it always holds that $m[g(x_2)-g(x_1)] > x_2 \cdot f(x_2) - x_1 \cdot f(x_1)$, find the range of the real number $m$.

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