\[\large \dfrac{(100!+99!)(99!-98!)(98!+97!)\cdots(2!+1!)}{(100!-99!)(99!+98!)(98!-97!)\cdots(2!-1!)}\]

The value of the above fraction can be written as mixed fraction \(a\dfrac{b}{c}\), where \(b\) and \(c\) & \(a\) and \(b\) are co-prime, but \(a\) and \(c\) are **not** co-prime. What is the product of \(a\), \(b\), and \(c\)?

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