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.