The set
\left\{(x, y) \in \mathbb{R}^{2} \mid\lfloor x+y\rfloor \cdot\lceil x+y\rceil=(\lfloor x\rfloor+\lceil y\rceil)(\lceil x\rceil+\lfloor y\rfloor), 0 \leq x, y \leq 100\right\}
can be thought of as a collection of line segments in the plane. If the total length of those line segments is a+b \sqrt{c} for c squarefree, find a+b+c. (\lfloor z\rfloor is the greatest integer less than or equal to z, and \lceil z\rceil is the least integer greater than or equal to z, for z \in \mathbb{R}.)