Is it possible to answer these queries using bitwise Gaussian Elimination in $$$O(\log{}(maxn))$$$ complexity?

• $$$add(x)$$$ — Add number $$$x$$$ to set $$$S$$$.
• $$$order(x)$$$ — Let, $$$X$$$ denote the sorted set of all possible xor-sums of elements from a subset of $$$S$$$. Find a maximum $$$k$$$ such that $$$x\ \ge\ k$$$-th element in $$$X$$$.