The Ackermann function is a computable function which grows very, very quickly as its inputs grow. For example, while \( A(1,2),\) \(A (2,2),\) and \(A(3,2) \) are equal to \(4,7,\) and \(29,\) respectively, \( A(4,2) \approx 2 \times 10^{19728} \).

The Ackermann function 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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