If $a, b, c, d$ are real numbers such that $a^2 + b^2 + c^2 + d^2 \leq 1$, find the maximum of the expression:
\[(a + b)^4 + (a + c)^4 + (a + d)^4 + (b + c)^4 + (b + d)^4 + (c + d)^4.\]

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