CF1566C MAX-MEX Cut

A binary string is a string that consists of characters $ 0 $ and $ 1 $ . A bi-table is a table that has exactly two rows of equal length, each being a binary string. Let $ \operatorname{MEX} $ of a bi-table be the smallest digit among $ 0 $ , $ 1 $ , or $ 2 $ that does not occur in the bi-table. For example, $ \operatorname{MEX} $ for $ \begin{bmatrix} 0011\\ 1010 \end{bmatrix} $ is $ 2 $ , because $ 0 $ and $ 1 $ occur in the bi-table at least once. $ \operatorname{MEX} $ for $ \begin{bmatrix} 111\\ 111 \end{bmatrix} $ is $ 0 $ , because $ 0 $ and $ 2 $ do not occur in the bi-table, and $ 0 < 2 $ . You are given a bi-table with $ n $ columns. You should cut it into any number of bi-tables (each consisting of consecutive columns) so that each column is in exactly one bi-table. It is possible to cut the bi-table into a single bi-table — the whole bi-table. What is the maximal sum of $ \operatorname{MEX} $ of all resulting bi-tables can be?

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In the first test case you can cut the bi-table as follows: - $ \begin{bmatrix} 0\\ 1 \end{bmatrix} $ , its $ \operatorname{MEX} $ is $ 2 $ . - $ \begin{bmatrix} 10\\ 10 \end{bmatrix} $ , its $ \operatorname{MEX} $ is $ 2 $ . - $ \begin{bmatrix} 1\\ 1 \end{bmatrix} $ , its $ \operatorname{MEX} $ is $ 0 $ . - $ \begin{bmatrix} 0\\ 1 \end{bmatrix} $ , its $ \operatorname{MEX} $ is $ 2 $ . - $ \begin{bmatrix} 0\\ 0 \end{bmatrix} $ , its $ \operatorname{MEX} $ is $ 1 $ . - $ \begin{bmatrix} 0\\ 0 \end{bmatrix} $ , its $ \operatorname{MEX} $ is $ 1 $ . The sum of $ \operatorname{MEX} $ is $ 8 $ .