Let $a,$ $b,$ $c$ be positive real numbers.  Find the minimum value of
\[\frac{\frac{1}{a} + \frac{1}{b} + \frac{1}{c}}{\frac{1}{a + b} + \frac{1}{a + c} + \frac{1}{b + c}}.\]
<|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
\[\frac{\frac{1}{a} + \frac{1}{b} + \frac{1}{c}}{\frac{1}{a + b} + \frac{1}{a + c} + \frac{1}{b + c}}.\]
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant

user
2
Let $a,$ $b,$ $c$ be positive real numbers.  Find the minimum value of
\[\frac{\frac{1}{a} + \frac{1}{b} + \frac{1}{c}}{\frac{1}{a + b} + \frac{1}{a + c} + \frac{1}{b + c}}.\]
train