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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