Contest Duration: - (local time) (120 minutes)
F - Square /

Time Limit: 2 sec / Memory Limit: 1024 MB

### 問題文

• 任意の 1\leq i < j\leq N に対し、(i,i) を左上のマスとし (j,j) を右下のマスとするような正方形領域に書き込まれた 1 の個数が偶数である

### 制約

• 2 \leq N \leq 10^5
• 0 \leq M \leq min(5 \times 10^4,N^2)
• 1 \leq a_i,b_i \leq N(1\leq i\leq M)
• 0 \leq c_i \leq 1(1\leq i\leq M)
• i \neq j ならば (a_i,b_i) \neq (a_j,b_j) である
• 入力はすべて整数である

### 入力

N M
a_1 b_1 c_1
:
a_M b_M c_M


### 入力例 1

3 3
1 1 1
3 1 0
2 3 1


### 出力例 1

8


101   111
011   111
000   011


### 入力例 2

4 5
1 3 1
2 4 0
2 3 1
4 2 1
4 4 1


### 出力例 2

32


### 入力例 3

3 5
1 3 1
3 3 0
3 1 0
2 3 1
3 2 1


### 出力例 3

0


### 入力例 4

4 8
1 1 1
1 2 0
3 2 1
1 4 0
2 1 1
1 3 0
3 4 1
4 4 1


### 出力例 4

4


### 入力例 5

100000 0


### 出力例 5

342016343


Score : 900 points

### Problem Statement

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).

An 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).

Takahashi 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.

• For all 1\leq i < j\leq N, there are even number of 1s in the square region whose top-left square is (i,i) and whose bottom-right square is (j,j).

### Constraints

• 2 \leq N \leq 10^5
• 0 \leq M \leq min(5 \times 10^4,N^2)
• 1 \leq a_i,b_i \leq N(1\leq i\leq M)
• 0 \leq c_i \leq 1(1\leq i\leq M)
• If i \neq j, then (a_i,b_i) \neq (a_j,b_j).
• All values in input are integers.

### Input

Input is given from Standard Input in the following format:

N M
a_1 b_1 c_1
:
a_M b_M c_M


### Output

Print the number of possible ways to write integers, modulo 998244353.

### Sample Input 1

3 3
1 1 1
3 1 0
2 3 1


### Sample Output 1

8


For example, the following ways to write integers satisfy the condition:

101   111
011   111
000   011


### Sample Input 2

4 5
1 3 1
2 4 0
2 3 1
4 2 1
4 4 1


### Sample Output 2

32


### Sample Input 3

3 5
1 3 1
3 3 0
3 1 0
2 3 1
3 2 1


### Sample Output 3

0


### Sample Input 4

4 8
1 1 1
1 2 0
3 2 1
1 4 0
2 1 1
1 3 0
3 4 1
4 4 1


### Sample Output 4

4


### Sample Input 5

100000 0


### Sample Output 5

342016343