
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 $f(x)=\ln x$ and $g(x)=\frac{1}{2}x|x|$.
(1) Find the equation of the tangent line to $g(x)$ at $x=-1$;
(2) Let $F(x)=x\cdot f(x)-g(x)$, find the monotonic intervals of $F(x)$;
(3) If for any $x\_1$, $x\_2\in[1,+\infty)$ and $x\_1 > x\_2$, the inequality $m[g(x\_1)-g(x\_2)] > x\_1f(x\_1)-x\_2f(x\_2)$ always holds, find the range of values for the real number $m$.

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