If 
$$\sum_{k=1}^{40} \left( \sqrt{1 + \frac{1}{k^{2}} + \frac{1}{(k + 1)^{2}}}\right) = a + \frac {b}{c}$$
where $a, b, c \in \mathbb{N}, b < c, \gcd(b,c) = 1,$ then what is the value of $a + b?$

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