Let $a, b, c$, and $d$ be prime numbers with $a \le b \le c \le d > 0$. 
Suppose $a^2 + 2b^2 + c^2 + 2d^2 = 2(ab + bc - cd + da)$. Find $4a + 3b + 2c + d$.

Let's think step by step and output the final answer within \boxed{}.
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