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I want a bound which only depends on $p$. That is, for any $0<a_0<p$, the bound is always correct. Your understanding and modification is really helpful. Thanks!
I mean the only $i$ such that $a_i=1$. In fact, I just wonder why such $i$ are always small, for any $p$ and $a_0$. I guess such $i$ is equal to $O(\log p)$, for example.