$(20$ points$)$ Design an $O(logn)-$time algorithm for the problem described below. $($i$)$ Provide your algorithm in pseudocode. $($ii$)$ Explain your algorithm in plain English discussing why it is correct, and $($iii$)$ justify why the running time is $O(logn).$

Find$-$Max $(A[1,$dots,$n])$ :

Input: An array $A[1,$dots,$n]$ of numbers with the following property. There exists $m$ such that $1mn$ and $A[1,$dots,$m]$ is a strictly increasing sequence and $A[m,$dots,$n]$ is a strictly decreasing sequence of numbers. That is$,$ there is some $m$ for which $A[1]A[m+1]>$cdots$>A[n]($For example, A could be the sequence $(1,5,8,12,11,4,2),$ where $m$ would be $4.)$

Output: i such that $A[i]=$max$\{A[1],$dots,$A[n]\}($In the given example, the correct output would be $4,$ since $A[4]=12$ is the largest number in the input array $A).$