After visiting the ancient mosaics in Paphos, an artist decides to create several black-and-white patterns. Each pattern should be a rectangular grid of $$$n$$$ rows and $$$m$$$ columns, satisfying the following properties.

The top and bottom boundaries of the grid are connected to each other, just as the left and right boundaries are, making a torus shape. Formally, let us index rows from $$$0$$$ to $$$n-1$$$ top to bottom, and columns from $$$0$$$ to $$$m-1$$$ left to right. A cell $$$(x, y)$$$ is at the intersection of row $$$x$$$ and column $$$y$$$. Then:

• cells $$$(x, y)$$$ and $$$((x + 1) \bmod n, y)$$$ are vertically adjacent,
• cells $$$(x, y)$$$ and $$$(x, (y + 1) \bmod m)$$$ are horizontally adjacent.

Each cell is painted either black or white. A painting is valid if all of the following conditions hold:

• No two vertically or horizontally adjacent cells are both black.
• For every cell $$$(i,j)$$$, at least one of the three vertically adjacent cells $$$(i,j)$$$, $$$((i+1) \bmod n,j)$$$, and $$$((i+2) \bmod n,j)$$$ is black.
• All white cells form one connected region: from any white cell, it is possible to reach any other by moving between adjacent white cells.

Count the number of different valid paintings modulo $$$998\,244\,353$$$. Two paintings are different if the colors of at least one cell differ.