Suppose $u$ modify a given sorted list of $4n$ distinct numbers as follows: Keep elements in even positions $($positions $2,4,6,..\mathrm{.}4$n$)$ as they are. Form $n$ disjoint pairs $(i,j)$ on the odd numbered positions where $i=2k+1$ for some $k=0$ to $n-1$ and $j=2k+1$ for some $k=n$ to $2n-1.$ Now swap elements in positions i and $j$ for every such pair. $($I$.$e$.$ every element in an odd numbered position in the first half of the array is swapped with some element in an odd numbered position in the second half of the array. No element is involved in more than one swap $($i$.$e$.$ the swaps are disjoint$).$ You don't know these $($i$,$ j$)$ pairs except that an element in an odd numbered position in the first half of the array is swapped with some element in an odd numbered position in the second half.

a$)$ Define the modification process for a given sorted list of $4n$ distinct numbers, involving keeping even$-$positioned elements unchanged and swapping elements in specified odd positions.

b$)$ Develop an algorithm to determine if a given element $x$ is present in the new reshuffled array, considering the swaps and the modified structure, with a time complexity of $O(logn).$

c$)$ Analyze the efficiency of the algorithm in terms of time complexity.

please also give pseudo code for b$)$