Compute sums and max path in N-ary tree

You are given the root of an N-ary tree (each node can have 0..k children). Implement the Node class yourself.

Assume each node stores an integer value and a list of child nodes children.

Complete the following tasks:

1. Sum of all nodes : Return the sum of value over all nodes in the tree.
2. Maximum path value : Define a path as a sequence of nodes from the root to any leaf (a node with no children). The path value is the sum of node values along that path. Return the maximum root-to-leaf path value.
3. Return the max path : Return the list of node values along one root-to-leaf path that achieves the maximum path value from (2). If multiple paths tie, you may return any one of them.

Input / Output

• Input: root: Node | null
• Output:
  • (1) an integer total sum
  • (2) an integer maximum root-to-leaf path sum
  • (3) an array/list of integers representing the values along the chosen max path

Notes / Constraints

• The tree can be empty ( root = null ).
• Node values can be negative, zero, or positive.
• Aim for linear time in the number of nodes.