Single Number III - Approach

$1.$ XOR all elements. Result is $a \oplus b$ where $a$ and $b$ are the two unique numbers.

$2.$ Find any set bit in this XOR (the rightmost is easiest: xor & (-xor)).

$3.$ This bit differs between $a$ and $b$. Partition all numbers by this bit.

$4.$ XOR within each group. One group gives $a$, the other gives $b$.