Binary search algorithm splits the set into $2$ sets of equal sizes $(\frac{1}{2}$ and $\frac{1}{2}).$ "MidtermSearch" algorithm is a variant of binary search algorithm, and it splits the set into $2$ sets of different sizes of one$-$third $(1/3)$ and two$-$thirds $(2/3)$ instead.

$($a$)$ Give the recurrence relations of "MidtermSearch" algorithm.