
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 functions $f\left(x\right)=3|x-1|+|3x+1|$ and $g\left(x\right)=|x+2|+|x-a|$, let the range of $f\left(x\right)$ be set $A$ and the range of $g\left(x\right)$ be set $B$.
$(1)$ Find $A$;
$(2)$ If $A\cup B=B$, find the range of the real number $a$.

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