Interstellar Intervals
It's the year 3000, and Bessie became the first cow in space! During her journey between the stars, she found a number line with N (2≤N≤5⋅105) points, numbered from 1 to N. All points are initially colored white. She can perform the following operation any number of times.
- Choose a position i within the number line and a positive integer x. Then, color all the points in the interval [i,i+x−1] red and all points in [i+x,i+2x−1] blue. All chosen intervals must be disjoint (i.e. no points in [i,i+2x−1] can be already colored red or blue). The entire interval must also fall within the number line (i.e. 1≤i≤i+2x−1≤N).
Farmer John gives Bessie a string s of length N consisting of characters R, B, and X. The string represents Farmer John's color preferences for each point: si=R means the i'th point must be colored red, si=B means the i'th point must be colored blue, and si=X means there is no constraint on the color for the i'th point.
Help Bessie count the number of distinct ways for the number line to be colored while satisfying Farmer John's preferences. Two colorings are different if there is at least one corresponding point with a different color. Because the answer may be large, output it modulo 109+7.
Bessie can choose i=1,x=1 (i.e. color point 1 red and point 2 blue) and i=3,x=2 (i.e. color points 3,4 red and points 5,6 blue) to produce the coloring RBRRBB.
The other colorings are RRBBRB, RBWWRB, RRRBBB, and RBRBRB.
The six colorings are WWRBWW, WWRBRB, WRRBBW, RBRBWW, RBRBRB, and RRRBBB.
SCORING:
- Input 4: N≤500
- Inputs 5-6: N≤104
- Inputs 7-13: All but at most 100 characters in s are X.
- Inputs 14-23: No additional constraints
Problem credits: Chongtian Ma, Alex Liang
Source: USA Computing Olympiad.
Input Format
The first line contains an integer N.
The following line contains string s.
Output Format
Output the number of distinct ways for the number line to be colored while satisfying Farmer John's preferences modulo 109+7.