Binary Reduction to Zero

You are given a non-negative integer n. Your objective is to transform this number into 0 by performing a carefully defined set of bit manipulation operations. You may execute these operations any number of times, in any order you choose.

The allowed operations are as follows:

• Operation 1 (Toggle Least Significant Bit): You may flip (change from 0 to 1, or from 1 to 0) the rightmost bit (the 0ᵗʰ position) of the binary representation of n at any time, unconditionally.

• Operation 2 (Conditional Higher Bit Toggle): For any bit position i (where i ≥ 1), you may flip the bit at the iᵗʰ position only if the following conditions are simultaneously satisfied:

  • The bit at position (i - 1) is currently 1.
  • All bits at positions (i - 2), (i - 3), ..., down to position 0 are all currently 0.

  In other words, you can toggle the iᵗʰ bit only when the bits to its right form the pattern: exactly one 1 at position (i - 1), followed by all 0s in positions (i - 2) through 0.

Your task is to determine the minimum number of operations required to reduce the given integer n to 0.

This problem is deeply connected to Gray code sequences, where consecutive numbers differ by exactly one bit. Understanding the mathematical structure behind Gray code and its inverse can lead to elegant closed-form solutions.