如图,在平面直角坐标系$xOy$中,点$C_{1}$,$C_{2}$,$\ldots$,都在$y$轴正半轴上,点$A_{1}$在二次函数$y=x^{2}(x \gt 0)$图象上,以$OA_{1}$,$OC_{1}$为邻边作平行四边形$OA_{1}B_{1}C_{1}$,且$OC_{1}=\sqrt{2}OA_{1}$,延长$C_{1}B_{1}$与二次函数$y=x^{2}(x \gt 0)$图象交于点$A_{2}$；以$C_{1}A_{2}$,$C_{1}C_{2}$为邻边作平行四边形$C_{1}A_{2}B_{2}C_{2}$,且$C_{1}C_{2}=\sqrt{2}C_{1}A_{2}$,延长$C_{2}B_{2}$与二次函数$y=x^{2}(x \gt 0)$图象交于点$A_{3}$；$\ldots$；按此规律进行下去,若$A_{1}$的横坐标为$1$,则$A_{2022}$的坐标为（ ）