Archit needs to go home!

Starting at \((2,2) \), an object moves in the coordinate plane via a sequence of steps, each of length 1. Each step is left, right, up or down, all four being equally likely. If the probability that the object reaches to \((4,4) \) in 6 or fewer steps is \( \dfrac a{64} \), then what is the value of \(a\)?

Note that the favorable path for finding the probability are those in which object stops as soon as it reaches \((4,4) \).

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