solution: By AM-GM,
  \[a + b + c + d \ge 4 \sqrt[4]{abcd},\]and
  \[\frac{1}{a} + \frac{1}{b} + \frac{1}{c} + \frac{1}{d} \ge 4 \sqrt[4]{\frac{1}{abcd}},\]so
  \[(a + b + c + d) \left( \frac{1}{a} + \frac{1}{b} + \frac{1}{c} + \frac{1}{d} \right) \ge 4 \sqrt[4]{abcd} \cdot 4 \sqrt[4]{\frac{1}{abcd}} = 16.\]Equality occurs when $a = b = c = d,$ so the minimum value is $\boxed{16}.$
prompt.content: <|im_start|>system
  You are a helpful assistant.<|im_end|>
  <|im_start|>user
  Let $a,$ $b,$ $c,$ $d$ be positive real numbers.  Find the minimum value of
  \[(a + b + c + d) \left( \frac{1}{a} + \frac{1}{b} + \frac{1}{c} + \frac{1}{d} \right).\]
  Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
  <|im_start|>assistant
  
prompt.role: user
extra_info.answer: 16
extra_info.question: Let $a,$ $b,$ $c,$ $d$ be positive real numbers.  Find the minimum value of
  \[(a + b + c + d) \left( \frac{1}{a} + \frac{1}{b} + \frac{1}{c} + \frac{1}{d} \right).\]
extra_info.split: test