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A graph is said to be -distinguishable if there is a -coloring of which no non-trivial automorphism preserves. That is, It was conjectured that if and the action on has no singleton orbits, then is 2-distinguishable. We give an example where this fails. We partially repair the conjecture by showing that when" enough motion occurs," the distinguishing number does indeed decay. Specifically, defining we show that when , is 2-distinguishable. In general, we show that if then is -distinguishable.