
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 $f(x)=3|x-a|+|3x+1|$ and $g(x)=|4x-1|-|x+2|$.
$(1)$ 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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