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 $(4,4)$ .