Let $a$, $b$, $c$, and $d$ be prime numbers such that $a > 3b > 6c > 12d$ and $a^2 - b^2 + c^2 - d^2 = 1749$. Determine all possible values of $a^2 + b^2 + c^2 + d^2$.

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