• # For Solution

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? Rubik’s Cube Coloring (hard version) solution codeforces

You have a perfect binary tree of 2k12k−1 nodes — a binary tree where all vertices ii from 11 to 2k112k−1−1 have exactly two children: vertices 2i2i and 2i+12i+1. Vertices from 2k12k−1 to 2k12k−1 don’t have any children. You want to color its vertices with the 66 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 nn 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 109+7109+7.

### Rubik’s Cube Coloring (hard version) solution codeforces

The first line contains the integers kk (1k601≤k≤60) — the number of levels in the perfect binary tree you need to color.

The second line contains the integer nn (1nmin(2k1,2000)1≤n≤min(2k−1,2000)) — the number of nodes, colors of which are already chosen.

The next nn lines contains integer vv (1v2k11≤v≤2k−1) and string ss — the index of the node and the color of the node (ss is one of the whiteyellowgreenbluered and orange).

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

### Rubik’s Cube Coloring (hard version) solution codeforces

Print one integer — the number of the different colorings modulo 109+7109+7.

Examples
input

Copy
3
2
5 orange
2 white


### Rubik’s Cube Coloring (hard version) solution codeforces

Copy
1024

input

Copy
2
2
1 white
2 white


### Rubik’s Cube Coloring (hard version) solution codeforces

Copy
0

input

Copy
10
3
1 blue
4 red
5 orange


### Rubik’s Cube Coloring (hard version) solution codeforces

Copy
328925088

Note

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