Let A be a sorted array of numbers where every value except one appears exactly two times; the remaining value appears only one time. Here are some sample inputs to the problem:

A$=[11,11,18,18,19,19,20,20,25,25,29]$ A$=[1,1,2,3,3,6,6,8,8,10,10]$

$2-1$

Design an O$($logn$)$ algorithm for finding which value appears only once. Please provide $1)$ An intuitive explanation of the algorithm, $2)$ the pseudocode of your proposed solutions, $3)$ proof of correctness, and $4)$ an asymptotic analysis of the running time.