如图,在$\triangle ABC$中,分别以$A$,$B$为圆心,大于$\frac{1}{2}AB$的长为半径作弧,两弧相交于$D$,$E$两点,分别以$A$,$C$为圆心,大于$\frac{1}{2}AC$的长为半径作弧,两弧相交于$F$,$G$两点,且分别与$BC$相交于$M$,$N$两点,连接$AM$、$AN$,若$\angle MAN=50^{\circ}$,则$\angle BAC=\left(\ \ \right)$