$(1)$如图,在$\triangle ABC$和$\triangle ADE$中,$AB=AC$,$AD=AE$,且$\angle BAC=\angle DAE$,$B$、$D$、$E$三点共线,$AC$与$DE$交于点$F$.
①求证:$BD=EC$；
②如图$2$,若点$G$是$BE$中点,且$\angle DAF=\angle ABD+\angle AEF$,连接$AG$、$CD$,求证:$CD=2AG$；
$(2)$若$AB=AC=3\sqrt{2}$,$BC=6$,且$\angle BAC=90^{\circ}$,在直线$BC$上取一点$D$,使得$CD=2$,连接$AD$,过$A$作$AE\bot AD$,且$AE=AD$,使直线$AC$和$ED$交于$F$,则$S_{\triangle BDE}=\_\_\_\_\_\_.$
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