Knotty Marble Probability

Satvik, Agnishom and Aditya each have a bag containing \(4\) differently colored marbles. Each bag contains the same four colours. Now they play a game,

First, Satvik picks one marble at random from each of Agnishom and Aditya's bag and places them in his own bag.

Next, Agnishom picks one marble at random from each of Satvik and Aditya's bag and places them in his own bag.

Finally, Aditya picks one marble at random from each of Satvik and Agnishom's bag and places them in his own bag.

At the end of the process, they open their bags and are thrilled to see that each bag contains \(4\) differently coloured marbles !

The probability of this event occurring can be expressed as \(\dfrac{a}{b}\) where \(a\) and \(b\) are positive co-prime integers.

Find \(a+b\)

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