(Canadian Senior Mathematics Contest 2020, Part A, Question 4, CEMC - UWaterloo)

Let $\left \lfloor x \right \rfloor$ denote the greatest integer which is less than or equal to $x$. For example, $\left \lfloor \pi \right \rfloor=3$. $S$ is the integer equal to the sum of the first $100$ terms shown: $$S=\left \lfloor \pi \right \rfloor+\left \lfloor \pi+\frac{1}{100} \right \rfloor+\left \lfloor \pi+\frac{2}{100} \right \rfloor+\left \lfloor \pi+\frac{3}{100} \right \rfloor+\dots+\left \lfloor \pi+\frac{99}{100} \right \rfloor$$ What is the value of $S$?