
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
Let $f(x) = \ln x$ and $g(x) = \frac{1}{2}x|x|$.  
(1) Find the equation of the tangent line to $g(x)$ at $x=-1$.  
(2) Let $F(x) = xf(x) - g(x)$, determine the intervals of monotonicity for $F(x)$.  
(3) If for all $x_1, x_2 \in [1, +\infty)$ with $x_1 > x_2$, it always holds that $m[g(x_1) - g(x_2)] > x_1f(x_1) - x_2f(x_2)$, find the range of values for the real number $m$.

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