Statement

Alex and Andrei are playing a game on an N × M matrix filled with non-negative integers. The rows of the matrix are numbered from 1 to N (from top to bottom), and the columns are numbered from 1 to M (from left to right). Alex is allowed to perform the following operation any number of times: (1) Choose a horizontal or vertical 1 × 3 subrectangle. (2) Select an integer V. (3) Add V to all numbers in the chosen subrectangle, ensuring that no number becomes negative. Andrei believes that Alex cannot make all the numbers in the matrix zero within at most 2 * N * M operations. Alex is determined to prove him wrong -- but since he hasn't learned addition at school yet, he needs your help. Can you determine whether Alex can reduce the entire matrix to zero within the given limit?