
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
(Elective 4-5: Special Topics on Inequalities)

Given the functions $f(x)=3|x-a|+|3x+1|, g(x)=|4x-1|-|x+2|$.

(1) Find the solution set of the inequality $g(x) < 6$;

(2) If there exist $x_1, x_2 \in \mathbb{R}$ such that $f(x_1)$ and $g(x_2)$ are opposite numbers, find the range of values for $a$.

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