SMO 2007 Open Round 1 Question 20

Suppose $0\lt a\lt b\lt c\lt d=2a$ and
$$(d-a)\left(\frac{a^2}{b-a}+\frac{b^2}{c-b}+\frac{c^2}{d-c}\right)=(a+b+c)^2.$$
Find $bcd/a^3$.