{"problem":{"name":"Square","description":{"content":"Takahashi has an $N \\times N$ grid. The square at the $i$\\-th row and the $j$\\-th column of the grid is denoted by $(i,j)$. Particularly, the top-left square of the grid is $(1,1)$, and the bottom-rig","description_type":"Markdown"},"platform":"AtCoder","limit":{"time_limit":2000,"memory_limit":262144},"difficulty":"None","is_remote":true,"is_sync":true,"sync_url":null,"sign":"caddi2018_d"},"statements":[{"statement_type":"Markdown","content":"Takahashi has an $N \\times N$ grid. The square at the $i$\\-th row and the $j$\\-th column of the grid is denoted by $(i,j)$. Particularly, the top-left square of the grid is $(1,1)$, and the bottom-right square is $(N,N)$.\nAn integer, $0$ or $1$, is written on $M$ of the squares in the Takahashi's grid. Three integers $a_i,b_i$ and $c_i$ describe the $i$\\-th of those squares with integers written on them: the integer $c_i$ is written on the square $(a_i,b_i)$.\nTakahashi decides to write an integer, $0$ or $1$, on each of the remaining squares so that the condition below is satisfied. Find the number of such ways to write integers, modulo $998244353$.\n\n*   For all $1\\leq i < j\\leq N$, there are even number of $1$s in the square region whose top-left square is $(i,i)$ and whose bottom-right square is $(j,j)$.\n\n## Constraints\n\n*   $2 \\leq N \\leq 10^5$\n*   $0 \\leq M \\leq min(5 \\times 10^4,N^2)$\n*   $1 \\leq a_i,b_i \\leq N(1\\leq i\\leq M)$\n*   $0 \\leq c_i \\leq 1(1\\leq i\\leq M)$\n*   If $i \\neq j$, then $(a_i,b_i) \\neq (a_j,b_j)$.\n*   All values in input are integers.\n\n## Input\n\nInput is given from Standard Input in the following format:\n\n$N$ $M$\n$a_1$ $b_1$ $c_1$\n$:$\n$a_M$ $b_M$ $c_M$\n\n[samples]","is_translate":false,"language":"English"}],"meta":{"iden":"caddi2018_d","tags":[],"sample_group":[["3 3\n1 1 1\n3 1 0\n2 3 1","8\n\nFor example, the following ways to write integers satisfy the condition:\n\n101   111\n011   111\n000   011"],["4 5\n1 3 1\n2 4 0\n2 3 1\n4 2 1\n4 4 1","32"],["3 5\n1 3 1\n3 3 0\n3 1 0\n2 3 1\n3 2 1","0"],["4 8\n1 1 1\n1 2 0\n3 2 1\n1 4 0\n2 1 1\n1 3 0\n3 4 1\n4 4 1","4"],["100000 0","342016343"]],"created_at":"2026-03-03 11:01:13"}}