
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 \(2x + 3y + z = 3\), and \(x \ge -2\), \(y \ge -\frac{2}{3}\), \(z \ge -4\). Find the maximum value of
\[
\sqrt{6x + 4} + \sqrt{9y + 2} + \sqrt{3z + 12}.
\]

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