In Prime Land, there are seven major cities, labeled $C_0$, $C_1$, \dots, $C_6$. For convenience, we let $C_{n+7} = C_n$ for each $n=0,1,\dots,6$; i.e., we take the indices modulo $7$. Al initially starts at city $C_0$.

Each minute for ten minutes, Al flips a fair coin. If the coin lands heads, and he is at city $C_k$, he moves to city $C_{2k}$; otherwise, he moves to city $C_{2k+1}$. If the probability that Al is back at city $C_0$ after $10$ moves is $\frac{m}{1024}$, find $m$.

Let's think step by step and output the final answer within \boxed{}.
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