P14794 [NERC 2025] Medical Parity

Nurse Mira works in an allergy clinic. For each patient Mira tests $n$ allergens in a fixed order. The outcome of the tests is written down as a binary string $x$ of length $n$: for each allergen, 1 means a positive reaction and 0 means no reaction. To analyze how the reactions are distributed, Mira also writes a $\emph{parity control string}$ for $x$. For a binary string $x$ of length $n$, the parity control string $y$ is defined as follows. For every position $i$ ($1 \le i \le n$), let $c_i$ be the number of characters equal to 1 among the first $i$ characters of $x$ (including position $i$). The parity control string $y$ is the binary string of length $n$ such that $y_i = c_i \mod 2$ for all $i$ ($1 \le i \le n$). In other words, $y_i$ is 1 if $c_i$ is odd and 0 if $c_i$ is even. For example, if $x = 11101$, then $y = 10110$. Unfortunately, when recording the data, some bits in the test result string and the parity control string may have been written incorrectly. For a given patient, Mira later finds in the system two binary strings $x'$ and $y'$ of the same length $n$. They were intended to be some true test result string $x$ and its parity control string $y$, but some bits in $x$ and $y$ might have been flipped during recording. For instance, in the previous example only the 3rd bit in $y$ could have been flipped, resulting in $x' = 11101$ and $y' = 10010$. In one $\emph{bit flip}$, a position in one of the two strings is chosen and the bit at this position is flipped (changing 0 to 1 or 1 to 0). Mira wants to know the minimal number of bit flips that could have happened when recording the data. Formally, you are given two binary strings $x'$ and $y'$ of length $n$. You want to obtain two strings $x$ and $y$ of length $n$ from $x'$ and $y'$ by flipping some bits in $x'$ and $y'$, so that $y$ is a parity control string of $x$. Find the minimal possible total number of bit flips needed.

The first line of the input contains the number of test cases $t$. The $2t$ lines follow --- two lines for each test case. The first line of each test case contains a non-empty binary string $x'$ consisting of characters 0 and 1. The second line contains a binary string $y'$ consisting of characters 0 and 1 with the same length as $x'$. The total length of all $x'$ strings in the input does not exceed $10^6$.

Print $t$ lines --- one line for each test case. For each test case, print a single integer --- the minimal possible number of bit flips that could have happened when recording the data.