P7524 The Witch’s Tea Party

“So that’s it. So this is the root of your desire. Interesting…” The voice echoed through the tomb. Subaru suddenly turned his head, but the scene in front of him changed at once: the dim walls around him became an endless lawn with hills of various sizes; and where the tomb entrance should have been, there was a tea table and two chairs. Under the shade of a parasol, there was also a black-clad woman sitting upright.  “I haven’t talked with humans for a long time, so I got a bit excited… My name is Echidna. The ‘Witch of Greed’—that should be easier to understand, right?” the woman said with a slight smile. “… Where is this!? I was supposed to be in a dark ruin. When did you move me here?” Subaru clenched his teeth and asked. “Unfortunately, you got it wrong. Your body wasn’t moved. You were only invited to my tea party.” “Tea party…?” “It’s the witch’s tea party.” **[Admin Mivik changed the timeline.]**

“Then… how long are you going to stand there? Sit down before the tea gets cold.” “Damn it… Fine, fine!” Subaru walked forward and sat opposite the witch. Besides two cups of tea, there was also a black-and-white chessboard on the table. “This is…?” Subaru pointed at the board and asked. “This is Witch Chess. I invited you here to answer a question that I have been thinking about for years but still can’t solve. If you solve it, I will give you and Emilia the right to take the trial… Otherwise…” An unsettling smile appeared on the witch’s face. “You might end up looping endlessly in the Sanctuary…?” Subaru broke out in a cold sweat and asked tremblingly, “So what is the question?” The witch picked up a piece beside her and said slowly: “As you can see, this is an $n\times n$ chessboard made of $n^2$ cells, and each cell can contain at most one piece. **Unlike a normal board, this board is cyclic. That is, the cell to the right of row $i$, column $n$ is row $i$, column $1$; the cell below row $n$, column $i$ is row $1$, column $i$; and so on.** The piece in my hand is called a ‘Sin Archbishop’. In one move, it can move any number of cells in any diagonal direction. The cells a piece can reach in one move are called the cells it controls. If one piece controls the cell where another piece is located, then the board configuration is unstable; otherwise, it is stable. Now, can you place exactly $m$ ‘Sin Archbishops’ on the board such that the whole configuration is stable?”  [High-resolution version of the figure above](https://cdn.luogu.com.cn/upload/image_hosting/np9zk0t4.png) Subaru thought for a while, then operated on the board and quickly completed the witch’s task, saying: “Then quickly give me the right to take the trial!” The witch said: “No… I still have one more question not solved…” Subaru: “Hey, you’re cheating!” The witch ignored Subaru and continued: “Now the question is: how many essentially different stable configurations are there such that there are exactly $m$ ‘Sin Archbishops’ in the configuration? Two configurations are essentially different if and only if there exists a cell that has a piece in one configuration but not in the other.” Subaru got even angrier, because he could not solve it. So he asked you for help. You only need to output the answer modulo $998244353$.

One line contains two positive integers $n,m$, representing the board size and the number of pieces you need to place.

One line contains one positive integer, representing the answer modulo $998244353$.

Sample 1: Placing one piece on a $1\times1$ board obviously has only one configuration. Sample 2: There are the following four stable configurations (use `.` for an empty cell, and `o` for a ‘Sin Archbishop’): ```plain oo .o .. o. .. .o oo o. ``` Sample 3: All $18$ stable configurations can be viewed [here](https://www.luogu.com.cn/paste/obkotkqb). ### Constraints For all data, $1\le n