Consider the sequence of natural numbers $a_n$ defined as $a_0=4$ and $a_{n+1}=\frac{a_n(a_n-1)}{2}$ for each $n\geq 0$. Define a new sequence $b_n$ as follows: $b_n=0$ if $a_n$ is even, and $b_n=1$ if $a_n$ is odd. Prove that for each natural $m$, the sequence \[b_m, b_{m+1}, b_{m+2},b_{m+3}, \dots\]is not periodic.