Collatz who? #1

The Collatz Procedure is $f=\text{If}\left[\text{EvenQ}[\text{\$\#\$1}],\frac{\text{\$\#\$1}}{2},3 \text{\$\#\$1}+1\right]\&$ or, in other words, one application of the procedure is if the number is even, then the result is division of the number by 2, otherwise the result of the procedure is multiply the number by 3 and then add 1 to the product. The procedure is normally applied to positive integers. In this problem, the procedure will only be applied to positive integers. Usually, if the procedure is applied to the previous result repetitively, then a 4-2-1 cycle will be reached. It is conceivable that some other cycle might be reached or the numbers will never reach a cycle of some sort.

This problem's question: Does any positive integer exist that does not enter a 4-2-1 cycle?

Note 1: You are not being asked to find such a positive integer.

Note 2: Consider this a research problem. Finding the answer on the web, in a library or so forth is acceptable.