将$2023$个边长为$1$的正方形按如图所示的方式排列,点$A$,$A_{1}$,$A_{2}$,$A_{3}$,$\ldots A_{2022}$和点$M$,$M_{1}$,$M_{2}$,$\ldots$,$M_{2021}$是正方形的顶点,连接$AM_{1}$,$AM_{2}$,$AM_{3}$,$\ldots AM_{2021}$分别交正方形的边$A_{1}M$,$A_{2}M_{1}$,$A_{3}M_{2}$,$\ldots A_{2022}M_{2021}$于点$N_{1}$,$N_{2}$,$N_{3}\ldots N_{2022}$,四边形$M_{1}N_{1}A_{1}A_{2}$的面积是$S_{1}$,四边形$M_{2}N_{2}A_{2}A_{3}$的面积是$S_{2}$,$\ldots$,则$S_{2022}$为（ ）