The right-most non-zero digit

It is a common exercise to determine the number of trailing zeros of a factorial. For example, \[ 20! = 2\; 432\; 902\; 008\; 176\; 64{\color{green}{0}}\; {\color{green}{000}} \] has 4 trailing zeros (as highlighted in green above).

However, finding the rightmost non-zero digit of a factorial is much harder. For example, the rightmost non-zero digit of \(20!\) is 4, as shown above.

Problem: Find the rightmost non-zero digit of \(10000!\).

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