Binary Tree Maximum Path Sum

Write `max_path_sum(root)` returning the largest sum along any path between two nodes. The path follows parent-child edges, need not touch the root, and must contain at least one node. Values may be negative. Same shape as Diameter: the recursion returns the best *downward* gain to its parent, while a running best records the path that turns at this node (`node.val + left + right`). Clamp a negative gain to zero — refusing a subtree is always allowed. Initialise the best to negative infinity, not zero, or an all-negative tree returns 0.