Suppose there exist constants $A$, $B$, $C$, and $D$ such that 
\[n^4 = A\binom{n}{4} + B\binom{n}{3} + C\binom{n}{2} + D\binom{n}{1}\]
 holds true for all positive integers $n \geq 4$. What is $A + B + C + D$?

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