CeilingAndFloor BasicSequence Intermediate

Problem - 3172
Let $S_n$ be the sum of first $n$ terms in sequence $\{a_n\}$ where $$a_n=\sqrt{1+\frac{1}{n^2}+\frac{1}{(n+1)^2}}$$ Find $\lfloor{S_n}\rfloor$ where the floor function $\lfloor{x}\rfloor$ returns the largest integer not exceeding $x$.

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