
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
If the solution set of the inequality $|ax-2| < 6$ is $\{x| -\frac{4}{3} < x < \frac{8}{3}\}$,  
$(1)$ find the value of $a$;  
$(2)$ if $b=1$, find the maximum value of $\sqrt{-at+12} + \sqrt{3bt}$.

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