如图,$\triangle ABC$中,$\angle C=90^{\circ}$,$AC=BC=a$,在$\triangle ABC$中截出一个正方形$A_{1}B_{1}C_{1}D_{1}$,使点$A_{1}$,$D_{1}$分别在$AC$,$BC$边上,边$B_{1}C_{1}$在$AB$边上；在$\triangle BC_{1}D_{1}$在截出第二个正方形$A_{2}B_{2}C_{2}D_{2}$,使点$A_{2}$,$D_{2}$分别在$BC_{1}$,$D_{1}C_{1}$边上,边$B_{2}C_{2}$在$BD_{1}$边上；$\ldots$,依此方法作下去,则:
$(1)$正方形$A_{1}B_{1}C_{1}D_{1}$的边长为____；
$(2)$第$n$个正方形的边长为____.