
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 $x + y + z = 2$, $x \ge -\frac{1}{2},$ $y \ge -2,$ and $z \ge -3$, with $2x + y = 1$. Find the maximum value of
\[
\sqrt{4x + 2} + \sqrt{3y + 6} + \sqrt{4z + 12}.
\]

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