Max-score grid descent with limited two-row jumps

You are given an N x M grid of integers board. Rows and columns are numbered from 0, and a cell's value is the score you collect when you visit it. Values may be negative.

A path starts at the top-row cell (0, p) and ends as soon as it reaches any cell in the bottom row (row N - 1). From cell (i, j) the path may move to:

• (i + 1, j - 1), (i + 1, j) or (i + 1, j + 1): one row down, shifting left, straight, or right.
• (i + 2, j): a jump straight down two rows. The cell (i + 1, j) in between is skipped and adds nothing.

Every move must land inside the grid. Over the whole path you may use at most K jumps. The score of a path is the sum of the cells it visits, including the start and the last cell. Two paths are different if their lists of visited cells differ.

Implement max_jump_path(board, p, K) that returns three things:

1. The highest possible path score.
2. One path with that score, written as (r,c)->(r,c)->.... If several paths tie, return the lexicographically smallest list of cells, comparing (row, col) pairs in order.
3. The number of distinct paths with that score, modulo 10^9 + 7.

Constraints: 1 <= N, M <= 500, 0 <= p < M, 0 <= K <= N, -10^4 <= board[i][j] <= 10^4.

Example 1

board = [[2,  0,  1],
         [-5, -9, -5],
         [4,  8,  3],
         [1,  2,  6]]
p = 1, K = 1

Output: 14, (0,1)->(2,1)->(3,2), 1. Jumping from (0,1) over the negative row to (2,1) and then stepping down-right gives 0 + 8 + 6 = 14.

Example 2

Same board, K = 0. Output: 9, (0,1)->(1,0)->(2,1)->(3,2), 2. Without a jump, the path must pass through row 1. Going through (1,0) or (1,2) both score 0 - 5 + 8 + 6 = 9, and (1,0) comes first.