Let $a$, $b$, $c$, and $d$ be real numbers such that \[ a^2 + 3b^2 + \frac{c^2 + 3d^2}{2} = a + b + c + d - 1. \] Find $1000a + 100b + 10c + d$.

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