Knitting Batik

Mr. Chanek wants to knit a batik, a traditional cloth from Indonesia. The cloth forms a grid $$$a$$$ with size $$$n \times m$$$. There are $$$k$$$ colors, and each cell in the grid can be one of the $$$k$$$ colors.

Define a sub-rectangle as an ordered pair of two cells $$$((x_1, y_1), (x_2, y_2))$$$, denoting the top-left cell and bottom-right cell (inclusively) of a sub-rectangle in $$$a$$$. Two sub-rectangles $$$((x_1, y_1), (x_2, y_2))$$$ and $$$((x_3, y_3), (x_4, y_4))$$$ have the same pattern if and only if the following holds:

• they have the same width ($$$x_2 - x_1 = x_4 - x_3$$$);
• they have the same height ($$$y_2 - y_1 = y_4 - y_3$$$);
• for every pair $$$(i, j)$$$ where $$$0 \leq i \leq x_2 - x_1$$$ and $$$0 \leq j \leq y_2 - y_1$$$, the color of cells $$$(x_1 + i, y_1 + j)$$$ and $$$(x_3 + i, y_3 + j)$$$ are equal.

Count the number of possible batik color combinations, such that the subrectangles $$$((a_x, a_y),(a_x + r - 1, a_y + c - 1))$$$ and $$$((b_x, b_y),(b_x + r - 1, b_y + c - 1))$$$ have the same pattern.

Output the answer modulo $$$10^9 + 7$$$.