For $k=1, 2, \ldots, 2011$, we denote $S_k = \frac{1}{k} + \frac{1}{k+1} + \cdots + \frac{1}{2011}$. Compute the sum $S_1 + S_1^2 + S_2^2 + \cdots + S_{2011}^2$.

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