Square roots and !'s!

Level pending

How many ordered pairs $$(a, b, c)$$ of nonnegative integers exist, satisfying

$\sqrt{a!} = b!\sqrt{c!}$

Where $$a, b,$$ and $$c$$ form an arithmetic progression with a constant difference of $$c$$?

NOTE: Unfortunately, this problem was posted with an incorrect answer. Because problems can now be edited but their answers cannot yet be edited, all I can do is write that whatever answer you get, you should add 1 to it.

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