Let $n$ be a fixed positive integer. For integer $m$ satisfying $|m| \le n$, define 
\[ S_m = \sum_{\substack{i-j=m \\ 0 \le i,j \le n}} \frac{1}{2^{i+j}}. \]
Then 
\[ \lim_{n\to \infty} \left( S_{-n}^2 + S_{-n+1}^2 + \ldots + S_n^2 \right) \]
can be expressed in the form $\frac{p}{q}$ for relatively prime positive integers $p, q$. Compute $100p + q$.

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