We are taking a random walk on the set of natural numbers. If we are at $M$, then with probability 1/4, we stay at $M$, with probability 5/12 we move to some random number less than or equal to $M/2$, and with probability 1/3, we move to a random greater than $M$ but less than or equal to $(3M+1)/2$. Is it true that almost every random walk like this ends in 0?