Climb Because The Mountain Is There

"Why climb? Because the mountain is there."

You are given a rooted tree with n nodes representing various points on Mushtagh Mountain. The nodes are numbered from 1 to n with node 1 being the mountain top. For each node i (2 ≤ i ≤ n), its parent is given as p_i. Define d_i as the number of edges from node i to the top (formally, d₁ = 0 and for 2 ≤ i ≤ n, d_i = d_pi + 1).

A climbing path is a procedure that starts from any node in {2, 3, …, n} and ends at the mountain top (node 1) through a sequence of moves. The corresponding climbing sequence is a sequence of nodes a₁, a₂, …, a_m with a₁ being the starting node and a_m = 1.

There are two types of moves allowed from any node i (2 ≤ i ≤ n):

1. Sprint: You are given a sprint range [l_i, r_i] for node i. In a sprint move, you choose an integer k with l_i ≤ k ≤ r_i and move upward exactly k steps, i.e. to the k-th ancestor of node i. It is guaranteed that 1 ≤ l_i ≤ r_i ≤ d_i.
2. Rest: Due to the treacherous terrain, when you rest you slide down to one of the children. Formally, you choose a node j satisfying p_j = i and move to node j. (If node i is a leaf, you cannot choose this move.)

In addition, for each node (either the starting node or one reached after a move) a parameter h_i is given with 0 ≤ h_i < d_i. For a climbing sequence a₁, a₂, …, a_m to represent a legal climbing path, the following must hold: For all indices 1 ≤ i < j ≤ m, if the move from a_i to a_j is a sprint move (i.e. it is not a rest move), then $$d_{a_j} < d_{a_i} - h_{a_i}.$$

Your task is to compute the total number of legal climbing paths starting from every node numbered from 2 through n. Two climbing paths are considered distinct if their climbing sequences differ. Since the answer can be very large, please output it modulo $$998\,244\,353$$.

inputFormat

The input begins with a single integer n denoting the number of nodes in the tree.

The second line contains n - 1 space-separated integers: p₂, p₃, …, p_n, where p_i is the parent of node i.

Then, for each node i from 2 to n, there is a line containing three integers: l_i, r_i, h_i.

outputFormat

Output a single integer: the total number of legal climbing paths starting from nodes 2 to n, taken modulo $$998\,244\,353$$.

sample

3
1 1
1 1 0
1 1 0

2