4 boxes, 8 balls

There are four boxes. Each contains 2 balls. The first box has a red and a white ball in it. The remaining three boxes each have two white balls in them.

A ball is picked at random from box 1 and put in box 2.
Then a ball is picked at random from box 2 and put into box 3.
Then a ball is picked at random from box 3 and put into box 4.
Finally, a ball is picked from box 4.

The probability that the ball picked from box 2 is red, given that the final ball picked from box 4 is white can be written as $\dfrac ab$, where $a$ and $b$ are coprime positive integers, find $a+b$.

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