
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
For each value of $x,$ $g(x)$ is defined to be the minimum value of the three numbers $3x + 3,$ $\frac{1}{3} x + 2,$ and $-\frac{1}{2} x + 8.$ Find the maximum value of $g(x).$

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