$2500$ chess kings have to be placed on a $100\times100$ so that
$(i)$no king can capture any other one (i.e. no two kings are plaaced in two squares sharing a common vertex);
$(ii)$ each row and each column contains exactly $25$ kings.
Find the number of such arrangements. (Two arrangements differifng by rotation or sym-metry are supposed to be different.)

Let's think step by step and output the final answer within \boxed{}.
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