Maximum-Score Grid Path with Limited Jumps

Given an integer grid, a top-row starting column, and a budget of at most K two-row jumps, maximize the score on a path to the bottom; then reconstruct an optimal path, count all optimal paths modulo 1e9+7, and incorporate adjacency-based bonuses. Requirements Input: an N x M integer grid board, a starting column p, and a jump budget K. Start at (0, p) and remain within the grid. From (i, j), move to (i+1, j-1), (i+1, j), (i+1, j+1), or use the special move (i+2, j). The special move may be used at most K times over the full path. The base score is the sum of all visited cell values. Part 1: return the maximum score obtainable when reaching the last row. Part 2: return one path that obtains the maximum score. Part 3: return the number of distinct maximum-score paths modulo 10^9 + 7. Part 4: incorporate two additional scoring rules: consecutive equal-valued cells add X, and three consecutively visited cells with strictly increasing values add Y. Examples board = [[1, 2, 3, 4], [5, 6, 1, 2], [7, 8, 9, 1], [3, 2, 5, 6]], p = 1, K = 1. One valid path is (0,1) -> (1,1) -> (2,2) -> (3,3). A valid path using the special jump is (0,1) -> (2,1) -> (3,2). Notes A two-row jump from row N-3 to the last row is valid; any move that would land outside the grid is invalid. For Part 4, the state must preserve enough recent path information to decide both the equal-pair and increasing-triple bonuses. Preparation Implement score maximization first, then add parent tracking without changing the score recurrence. Extend the state to count ties modulo 10^9 + 7 and test cases with several equal-score paths. Practice adding bonuses that depend on the previous one or two visited values while retaining the jump budget.