Carry out the following operations showing the resulting $2-3$ tree after each operation. Assume the tree is initially empty $($i$.$e$.$ the root is null$).$ After performing the operation, use the resulting tree for the next operation.

a$.$ insert$(45)$

b$.$ insert$(20)$

c$.$ insert$(30)$

d$.$ insert$(60)$

e$.$ insert$(80)$

f$.$ insert$(70)$