A sequence of integers, $\{a_{n}\}_{n \ge 1}$ with $a_{1}>0$, is defined by \[a_{n+1}=\frac{a_{n}}{2}\;\;\; \text{if}\;\; n \equiv 0 \;\; \pmod{4},\] \[a_{n+1}=3 a_{n}+1 \;\;\; \text{if}\;\; n \equiv 1 \; \pmod{4},\] \[a_{n+1}=2 a_{n}-1 \;\;\; \text{if}\;\; n \equiv 2 \; \pmod{4},\] \[a_{n+1}=\frac{a_{n}+1}{4}\;\;\; \text{if}\;\; n \equiv 3 \; \pmod{4}.\] Prove that there is an integer $m$ such that $a_{m}=1$.