You Don't See This Expression Every Day

Let x1,x2,x3x_1,x_2,x_3\ldots be the list of positive integers in strictly increasing order such that xx+x!x+1\dfrac{x^x+x!}{x+1} is an integer. Find x100x_{100}.

Details and Assumptions\text{Details and Assumptions}

This note by Arron Kau may help.

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