
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
The domain of the function $f(x) = \log_{3}(x^{2} + 2x - 8)$ is $A$, and the function $g(x) = x^{2} + (m+1)x + m$.  
(1) If $m = -4$, the solution set of $g(x) \leq 0$ is $B$, find $A \cap B$;  
(2) If there exists $x \in [0, \frac{1}{2}]$ such that the inequality $g(x) \leq -1$ holds, find the range of the real number $m$.

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