Rubik's Cube Coloring (hard version)
Contest: Codeforces Round 747 (Div. 2)
Difficulty rating: 2300
Tags: brute force, dp, implementation, math, trees

It is the hard version of the problem. The difference is that in this version, there are nodes with already chosen colors.

Theofanis is starving, and he wants to eat his favorite food, sheftalia. However, he should first finish his homework. Can you help him with this problem?

You have a perfect binary tree of $$$2^k - 1$$$ nodes — a binary tree where all vertices $$$i$$$ from $$$1$$$ to $$$2^{k - 1} - 1$$$ have exactly two children: vertices $$$2i$$$ and $$$2i + 1$$$. Vertices from $$$2^{k - 1}$$$ to $$$2^k - 1$$$ don't have any children. You want to color its vertices with the $$$6$$$ Rubik's cube colors (White, Green, Red, Blue, Orange and Yellow).

Let's call a coloring good when all edges connect nodes with colors that are neighboring sides in the Rubik's cube.

A picture of Rubik's cube and its 2D map.

More formally:

- a white node can not be neighboring with white and yellow nodes;
- a yellow node can not be neighboring with white and yellow nodes;
- a green node can not be neighboring with green and blue nodes;
- a blue node can not be neighboring with green and blue nodes;
- a red node can not be neighboring with red and orange nodes;
- an orange node can not be neighboring with red and orange nodes;

However, there are $$$n$$$ special nodes in the tree, colors of which are already chosen.

You want to calculate the number of the good colorings of the binary tree. Two colorings are considered different if at least one node is colored with a different color.

The answer may be too large, so output the answer modulo $$$10^9+7$$$.

Input format:
The first line contains the integers $$$k$$$ ($$$1 \le k \le 60$$$) — the number of levels in the perfect binary tree you need to color.

The second line contains the integer $$$n$$$ ($$$1 \le n \le \min(2^k - 1, 2000)$$$) — the number of nodes, colors of which are already chosen.

The next $$$n$$$ lines contains integer $$$v$$$ ($$$1 \le v \le 2^k - 1$$$) and string $$$s$$$ — the index of the node and the color of the node ($$$s$$$ is one of the white, yellow, green, blue, red and orange).

It is guaranteed that each node $$$v$$$ appears in the input at most once.

Output format:
Print one integer — the number of the different colorings modulo $$$10^9+7$$$.

Constraints:
Time limit: 3.0 s
Memory limit: 256.0 MB

Note:
In the picture below, you can see one of the correct colorings of the first test example.

Example input:
3
2
5 orange
2 white

Example output:
1024

Example input:
2
2
1 white
2 white

Example output:
0

Editorial:
Let's define a node as marked if it has a predefined node in its subtree. There is always at least $$$1$$$ marked node since all predefined nodes are definitely marked.
You can see that marked nodes form a path for the root to any predefined node. Thus there are at most $$$n \cdot k$$$ marked nodes and we can run a standard $$$dp[i][j]$$$ on them (node $$$i$$$ is colored with $$$j$$$). Depending on the implementation the $$$dp$$$ can have time complexity $$$O(n \cdot k \cdot 6 \cdot 4)$$$ or $$$O(n \cdot k \cdot log(n \cdot k) \cdot 6 \cdot 4)$$$ if you use map.
You multiply the result with $$$4^{m}$$$ where $$$m$$$ is the number of unmarked nodes. This holds because if their parent has a fixed color they always have $$$4$$$ choices and so on.