Harder than you think (3)

$\large \displaystyle 1^1 \times 2^2 \times 3^3 \times 4^4 \times\cdots \times 98^{98} \times 99^{99} \times 100^{100}$

$\,$

$\,$

Find the last five non-zero digits of the number above.

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Before you try this, try Problem 1 and Problem 2 .