$A(1,2),$ $A (2,2),$ and $A(3,2)$ are equal to $4,7,$ and $29,$ respectively, $A(4,2) \approx 2 \times 10^{19728}$.

The Ackermann function is a computable function which grows very, very quickly as its inputs grow. For example, whileThe Ackermann function can be defined as follows: $A(m,n) = \begin{cases} n+1 & \mbox{if } m = 0 \\ A(m-1, 1) & \mbox{if } m > 0 \mbox{ and } n = 0 \\ A(m-1, A(m, n-1)) & \mbox{if } m > 0 \mbox{ and } n > 0. \end{cases}$

What are the last 3 digits of $A(4,5) ?$

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