
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 $x \ge -\frac{1}{2}$, $y \ge -2$, and $z \ge -3$. Find the maximum value of
\[
\sqrt{6x + 3} + \sqrt{6y + 12} + \sqrt{6z + 18}.
\]

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