Polycarp owns a shop in the capital of Berland. Recently the criminal activity in the capital increased, so Polycarp is thinking about establishing some better security in the storehouse of his shop.
The storehouse can be represented as a matrix with
n rows and
m columns. Each element of the matrix is either
. (an empty space) or
x (a wall).
Polycarp wants to hire some guards (possibly zero) to watch for the storehouse. Each guard will be in some cell of matrix and will protect every cell to the right of his own cell and every cell to the bottom of his own cell, until the nearest wall. More formally, if the guard is standing in the cell
(x0,y0), then he protects cell
(x1,y1) if all these conditions are met:
-
(x1,y1) is an empty cell;
-
either x0=x1 and y0≤y1, or x0≤x1 and y0=y1;
-
there are no walls between cells (x0,y0) and (x1,y1). There can be a guard between these cells, guards can look through each other.
Guards can be placed only in empty cells (and can protect only empty cells). The
plan of placing the guards is some set of cells where guards will be placed (of course, two plans are different if there exists at least one cell that is included in the first plan, but not included in the second plan, or vice versa). Polycarp calls a plan
suitable if there is
not more than one empty cell that is not protected.
Polycarp wants to know the number of suitable plans. Since it can be very large, you have to output it modulo
109+7.
Note
In the first example you have to put at least one guard, so there are three possible arrangements: one guard in the cell
(1,1), one guard in the cell
(1,3), and two guards in both these cells.