Two Sum and Variants

A Delivery Hero India client processes compact arrays and encoded values. Implement one function that performs three independent analyses efficiently:

1. Two Sum: Return the indices of two distinct elements in nums whose values add to target. Return the first pair found while scanning from left to right, or [] if no pair exists.
2. Binary Sum: Add two non-negative binary strings binary_a and binary_b without converting either full string to an integer. Return the canonical binary result without leading zeroes, except that zero is represented as "0".
3. First Duplicate: Scan values from left to right and return the first value whose second occurrence appears earliest. Return None if every value is unique.

Return a dictionary with keys two_sum, binary_sum, and first_duplicate.

Formal Specification

nums is a list of integers, target is an integer, binary_a and binary_b are binary strings, and values is a list of hashable values. The result is a dictionary containing a list of indices, a binary string, and either a value or None.