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

user
\frac{1024}{27}
Let $a,$ $b,$ $c,$ $d$ be positive real numbers.  Find the minimum value of
\[(1 + a)(1 + b)(1 + c)(1 + d) \left( \frac{1}{a} + \frac{1}{b} + \frac{1}{c} + \frac{1}{d} \right).\]
train