Too much recursion

The Ackermann function is defined as

\[A(m,n) = \begin{cases} n+1 & \mbox{if } m =0 \\ A(m-1,1) & \mbox{if } m>0 \mbox{ and } n=0\\A(m-1,A(m,n-1)) & \mbox{if } m > 0 \mbox{ and } n>0. \end{cases} \]

For all integers \(n > 1\), the value of \(A(4,n)\) all have the same remainder when divided by 1000. What is this remainder?

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