Let $S=\{0,1\}^n$ be a binary string of length $n$. Suppose you pick a number $r$ at random from any distribution on $\{1,\ldots,R\}$ of your choice and randomly generate $r$ boolean hash functions $h_0, h_1,\ldots, h_{r-1}: \{0,1\}\to\{0,1\}$. Using these hash functions, you take a one-directional random walk on $S$, starting at time $t=1$ and index $i=1$ so that if $h_{t \bmod{r}}(S[i])=1$, then you move from index $i$ to $i+1$. In other words, if the value of the hash function, indexed by your current time, of the character of your current position is $1$, then you move forwards in $S$. The random walk ends when you've processed the entire string $S$, i.e., when you've hit index $n+1$.
Conditioned on the random walk ending, do there exist $R=\text{polylog}(n)$ and constant $c>0$ for which we can upper bound the probability that the random walk ends at time $t\equiv 0 \bmod{r}$ by at most $O\left(\frac{1}{R^c}\right)$? Note that for $R=O(1)$, the probability is bounded by $\frac{1}{2}$ depending on the realization of $h_{r-1}$ and for $R=\Omega(n)$ with sufficiently large constant, the probability is bounded by $O\left(\frac{1}{\sqrt{n}}\right)$ from anti-concentration.