Problem - Add Binary

Given two binary strings $a$ and $b$, return their sum as a binary string. For example, if $a = "1010"$ and $b = "1011"$, return $"10101"$.

You cannot convert to integers and add, because the inputs can be long (up to $10^4$ characters). You must perform binary addition manually, digit by digit, handling carries.

This is like adding two numbers on paper: start from the right, add digits and carry over when the sum exceeds the base. Before reading on, think about how to track the carry and build the result.