Let $1...$ be a sorted array of distinct integers, some of which may be negative. $($a$)$ Design a $($log $)$Divide and Conquer algorithm that finds an index such that $1$ and $=$ or correctly reports that such an index does not exist. Prove that your algorithm runs in $($log $)$ in all cases. $($b$)$ Suppose that we also know that $1=0.$ Design an even faster algorithm that either computes the index such that $=$ or correctly reports that such an index does not exist.