Let $a, b, c,$ and $d$ be real numbers such that:
\[
a + b = c + d + 12
\]
and
\[
ab + cd - 28 = bc + ad.
\]
Find the minimum possible value of $a^4 + b^4 + c^4 + d^4$.

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