Problem $5.$ Find the Missing Integer.

Suppose $k$ is a positive integer, and let $S$ be the set of integers from $0$ to $2^k-1($inclusive$).$ Then $|S|=2^k,$ and $S$ is the set of integers that can be represented as a k$-$bit binary integer. $($Recall: the bit sequence $b_{k-1}dots{b}_1{b}_0$ represents the integer $x={\displaystyle \underset{j=0}{\overset{k-1}{}}}{2}^j{b}_j,$ where each bit $b_j$ is either $0$ or $1.)$

Now suppose and you are given an array $A[1..n],$ where each $A[i]$ is in $S.$ Since there is at least one integer $x$ in $S$ that is not listed in $A.$ You want to write an algorithm that finds and returns such an $x.$ However, you do not have direct access to the array $A!$ Instead, for an array index $)i(n$ and bit index $j{j}^{th}A[i]A[i]kO(1)A[3](3,0)=0(3,1)=1$get$(3,2)=1xk,nO(n)nk=(nlgn)nkn(i,j)n=0x=0$