
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 \(a + b + c = 2\), and \(a \ge -\frac{1}{2}\), \(b \ge -1\), \(c \ge -2\). Find the maximum value of
\[
\sqrt{4a + 2} + \sqrt{4b + 4} + \sqrt{4c + 8}.
\]

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