Almost a Pythagorean Triple

Number Theory Level pending

You're given a right triangle as shown above. If \(a\) is a positive integer such that it satisfies the constraints \(a|c\) and \(b^{2} = a! + (a-1)! + (a-2)!\), what is the largest known value of \(a\)?

Hint: The related problem remains one of the world's unsolved math mysteries.

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