
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 \( a + b + c = 3 \), \( a \geq -1 \), \( b \geq -2 \), and \( c \geq -4 \). Find the maximum value of:
\[ \sqrt{4a + 4} + \sqrt{4b + 8} + \sqrt{4c + 16} \]
given that \( abc \geq 0 \).

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