
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
Let $x,$ $y,$ $z$ be real numbers such that $x + y + z = 2,$ and $x \ge -\frac{1}{2},$ $y \ge -\frac{3}{2},$ and $z \ge -1.$ Find the maximum value of 
\[\sqrt{3x + 1.5} + \sqrt{3y + 4.5} + \sqrt{3z + 3}.\]

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