CF2234D XOR, Expression and Two Binary Numbers

You are given an integer $ k $ . There is a sequence of $ n $ -bit binary numbers $ a_1, a_2, \ldots, a_{2^k + 1} $ . The numbers $ a_1 $ and $ a_{2^k + 1} $ are given to you, and the others are unknown. Then the unknown numbers are filled in as follows over $ k $ steps: - Suppose that before the $ i $ -th step, the numbers with indices $ p_1 \lt p_2 \lt \ldots \lt p_m $ have already been filled. (Before the first step, these are the numbers with indices $ 1, 2^{k} + 1 $ ). - Then for each $ j $ from $ 1 $ to $ m - 1 $ , the assignment $ a_{\frac{p_j + p_{j + 1}}{2}} := a_{p_j} \oplus a_{p_{j + 1}} $ $ ^{\text{∗}} $ is performed. - These assignments happen simultaneously, and after that all these numbers also become filled. It can be shown that this process always fills all numbers completely. This is what the process looks like for $ k = 2 $ and $ n = 3 $ , where initially $ a_1 = \texttt{010}, a_5 = \texttt{110} $ : - Before the first step, the numbers with indices $ 1, 5 $ are filled. Therefore, this operation will perform $ a_3 := a_1 \oplus a_5 = \texttt{010} \oplus \texttt{110} = \texttt{100} $ . - Before the second step, the numbers with indices $ 1, 3, 5 $ are filled. Therefore, this operation will perform $ a_2 := a_1 \oplus a_3 = \texttt{110} $ and $ a_4 = a_{3} \oplus a_{5} = \texttt{010} $ You need to compute the following expression: $ x_1 \cdot y_1 + x_2 \cdot y_2 + \ldots + x_{2^k + 1} \cdot y_{2^k + 1} $ , where $ x_i $ is the number of set bits in the $ i $ -th number, and $ y_i $ is the number of zero bits in the $ i $ -th number. $ ^{\text{∗}} $ $ x \oplus y $ denotes the bitwise exclusive OR of numbers $ x $ and $ y $

Each test contains multiple test cases. The first line contains the number of test cases $ t $ ( $ 1 \le t \le 10^4 $ ). The description of the test cases follows. The first line of each test case contains two integers $ n, k $ ( $ 1 \leq n \leq 10^5 $ , $ 1 \leq k \leq 30 $ ) — the length of the binary numbers and the number determining the number of binary numbers in the sequence. The second line of each test case contains a binary string $ s $ of length $ n $ ( $ s_i \in \{\texttt{0}, \texttt{1}\} $ ) — the value of $ a_1 $ . The third line of each test case contains a binary string $ z $ of length $ n $ ( $ z_i \in \{\texttt{0}, \texttt{1}\} $ ) — the value of $ a_{2^k + 1} $ . It is guaranteed that the sum of $ n $ over all test cases does not exceed $ 10^5 $ .

For each test case, output one integer — the value of the expression from the statement.

In the first test case, the process was described in the statement. The resulting sequence of binary numbers is $ [\texttt{010}, \texttt{110}, \texttt{100}, \texttt{010}, \texttt{110}] $ . Then the expression in the statement equals $ 1 \cdot 2 + 2 \cdot 1 + 1 \cdot 2 + 1 \cdot 2 + 2 \cdot 1 = 10 $ . In the second test case, at the first step we have $ a_2 = a_1 \oplus a_3 = \texttt{0} \oplus \texttt{0} = \texttt{0} $ . Therefore, the resulting sequence of numbers is $ [\texttt{0}, \texttt{0}, \texttt{0}] $ . For it, the value of the expression is zero.