
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 $a,$ $b,$ $c$ be real numbers such that $a + b + c = 3,$ and $a \ge -\frac{1}{2},$ $b \ge -\frac{3}{2},$ and $c \ge -2.$  Find the maximum value of
\[\sqrt{4a + 2} + \sqrt{4b + 6} + \sqrt{4c + 8}.\]

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