Given that $a + b + c = 5$ and $1 \le a,$ $b,$ $c \le 2,$ find the minimum value of
\[\frac{1}{a + b} + \frac{1}{b + c}.\]
<|im_start|>system
You are a helpful assistant.<|im_end|>
<|im_start|>user
Given that $a + b + c = 5$ and $1 \le a,$ $b,$ $c \le 2,$ find the minimum value of
\[\frac{1}{a + b} + \frac{1}{b + c}.\]
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant

user
\frac{4}{7}
Given that $a + b + c = 5$ and $1 \le a,$ $b,$ $c \le 2,$ find the minimum value of
\[\frac{1}{a + b} + \frac{1}{b + c}.\]
train