Let $a,$ $b,$ $c$ be positive real numbers.  Find the minimum value of
\[\left( 2a + \frac{1}{3b} \right)^2 + \left( 2b + \frac{1}{3c} \right)^2 + \left( 2c + \frac{1}{3a} \right)^2.\]
<|im_start|>system
You are a helpful assistant.<|im_end|>
<|im_start|>user
Let $a,$ $b,$ $c$ be positive real numbers.  Find the minimum value of
\[\left( 2a + \frac{1}{3b} \right)^2 + \left( 2b + \frac{1}{3c} \right)^2 + \left( 2c + \frac{1}{3a} \right)^2.\]
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant

user
8
Let $a,$ $b,$ $c$ be positive real numbers.  Find the minimum value of
\[\left( 2a + \frac{1}{3b} \right)^2 + \left( 2b + \frac{1}{3c} \right)^2 + \left( 2c + \frac{1}{3a} \right)^2.\]
train