In $\triangle A B C$, if $C=120^{\circ}, c=\sqrt{19}$ and $b=3$, then $a=$

In a $\triangle A B C, 2 A+C=300^{\circ}$. If the circumradius of the $\triangle A B C$ is eight times its inradius, then $\sin \frac{C}{2}=$

In $\triangle A B C$, if $a=5, b=4$ and $\cos (A-B)=\frac{31}{32}$, then $c=$

In $\triangle A B C$, if $A, B, C$ are in arithmetic progression, then

$$ \sqrt{a^2-a c+c^2} \cdot \cos \left(\frac{A-C}{2}\right)= $$