Question $1$: $k-$th Smallest Element from Two Lists $(8$ points$)$

Suppose you are given two sorted lists $A[1,$dots,$n]$ and $B=[1,$dots,$m]$ of size $n$ and $m,$ respectively.

Give an $O(logk)$ algorithm to find the $k-$th smallest element in $AB,$ i$.$e$.,$ of the combination of the two

arrays. For simplicity, you can assume $k$min$(m,n).$ Justify the correctness and running time of your

algorithm. $($Hint: What other algorithms do you know that have logarithmic running time? Can a similar

approach be used here?$)$