US-TST 2023 Problem 4

For nonnegative integers $a$ and $b$, denote their bitwise xor by $a \oplus b$. (For example, $9 \oplus 10=1001_{2} \oplus 1010_{2}=0011_{2}=3$.) Find all positive integers $a$ such that for any integers $x>y \geq 0$, we have $$x \oplus a x \neq y \oplus a y .$$