For any real number $x,$ let $\left[x\right]$ denote the integer part of $x\mathit{;}\mathit{ }\left\{x\right\}$ be the fractional part of $x\mathit{.}$ $\left(\left\{x\right\}=x-\left[x\right]\right)$. Let $A$ denote the set of all real numbers $x$ satisfying $\left\{x\right\}=\frac{x+\left[x\right]+\left[x+\left({\displaystyle \frac{1}{2}}\right)\right]}{20}$. If $S$ is the sum of all numbers in $A,$ find $\left[S\right]$.