The people in an infinitely long line are numbered $1,2,3,\dots$. Then, each person says either "Karl" or "Lark" independently and at random. Let $S$ be the set of all positive integers $i$ such that people $i$, $i+1$, and $i+2$ all say "Karl," and define $X = \sum_{i \in S} 2^{-i}$. Then the expected value of $X^2$ is $\frac{m}{n}$, where $m$ and $n$ are positive integers with $\gcd(m,n)=1$. Compute $100m+n$.

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