Problem:
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

Solution in C# (Mono 4.6.2.0) (judge verdict: Accepted)

using System;
using System.Collections.Generic;
using System.Linq;
using System.Linq.Expressions;
using System.IO;

using static util;
using P = pair<int, int>;

using Binary = System.Func<System.Linq.Expressions.ParameterExpression, System.Linq.Expressions.ParameterExpression,
                           System.Linq.Expressions.BinaryExpression>;
using Unary = System.Func<System.Linq.Expressions.ParameterExpression, System.Linq.Expressions.UnaryExpression>;

class Program
{
    static StreamWriter sw = new StreamWriter(Console.OpenStandardOutput()) { AutoFlush = false };
    static Scan sc = new Scan();
    const int M = 1000000007;
    const int M2 = 998244353;
    const long LM = (long)1e18;
    const double eps = 1e-11;
    static readonly int[] dd = { 0, 1, 0, -1, 0 };
    const string dstring = "RDLU";
    static void Main()
    {
        int n, m;
        sc.Multi(out n, out m);
        var dic = new SortedDictionary<int, int>[n];
        for (int i = 0; i < n; i++)
        {
            dic[i] = new SortedDictionary<int, int>();
        }
        int cc = 0;
        var val = new int[5][];
        for (int i = 0; i < 5; i++)
        {
            val[i] = new int[n];
            for (int j = 0; j < n; j++)
            {
                val[i][j] = -1;
            }
        }
        for (int i = 0; i < m; i++)
        {
            int a, b, c;
            sc.Multi(out a, out b, out c);
            --a;
            --b;
            if (Math.Abs(a - b) >= 3) {
                if (dic[b].ContainsKey(a)) {
                    if (dic[b][a] != c) {
                        DBG(0);
                        return;
                    }
                }
                else {
                    ++cc;
                    dic[a].Add(b, c);
                }
            }
            else {
                int k = a - b + 2;
                val[k][Math.Min(a, b)] = c;
            }
        }
        var dp = new long[16];
        for (int i = 0; i < 16; i++)
        {
            if (val[2][0] != -1 && val[2][0] != (i & 1)) continue;
            if (val[3][0] != -1 && val[3][0] != ((i >> 1) & 1)) continue;
            if (val[1][0] != -1 && val[1][0] != ((i >> 2) & 1)) continue;
            if (val[2][1] != -1 && val[2][1] != ((i >> 3) & 1)) continue;
            int c = 0;
            for (int j = 0; j < 4; j++)
            {
                if (((i >> j) & 1) == 1) {
                    ++c;
                }
            }
            if (c % 2 > 0) continue;
            dp[i] = 1;
        }
        for (int i = 0; i + 2 < n; i++)
        {
            var nex = new long[16];
            for (int l = 0; l < 16; l++)
            {
                for (int j = 0; j < 32; j++)
                {
                    if (val[4][i] != -1 && val[4][i] != (j & 1)) continue;
                    if (val[3][i + 1] != -1 && val[3][i + 1] != ((j >> 1) & 1)) continue;
                    if (val[2][i + 2] != -1 && val[2][i + 2] != ((j >> 2) & 1)) continue;
                    if (val[1][i + 1] != -1 && val[1][i + 1] != ((j >> 3) & 1)) continue;
                    if (val[0][i] != -1 && val[0][i] != ((j >> 4) & 1)) continue;
                    int c = 0;
                    for (int k = 0; k < 5; k++)
                    {
                        if (((j >> k) & 1) == 1) ++c;
                    }
                    if (c % 2 > 0) continue;
                    c = ((l >> 3) & 1);
                    if (((j >> 1) & 1) == 1) ++c;
                    if (((j >> 2) & 1) == 1) ++c;
                    if (((j >> 3) & 1) == 1) ++c;
                    if (c % 2 > 0) continue;
                    int ne = (l >> 3) & 1;
                    if (((j >> 1) & 1) == 1) ne |= 2;
                    if (((j >> 2) & 1) == 1) ne |= 8;
                    if (((j >> 3) & 1) == 1) ne |= 4;
                    nex[ne] = (nex[ne] + dp[l]) % M2;
                }
            }
            dp = nex;
        }
        long rem = (n * (long)n - ((n - 2) * 5 + 4) - cc * 2) / 2;
        MyMath.Mod = M2;
        Prt(dp.Sum() % M2 * MyMath.pow(2, rem) % M2);

        sw.Flush();
    }

    static void DBG(string a) => Console.WriteLine(a);
    static void DBG<T>(IEnumerable<T> a) => DBG(string.Join(" ", a));
    static void DBG(params object[] a) => DBG(string.Join(" ", a));
    static void Prt(string a) => sw.WriteLine(a);
    static void Prt<T>(IEnumerable<T> a) => Prt(string.Join(" ", a));
    static void Prt(params object[] a) => Prt(string.Join(" ", a));
}
class pair<T, U> : IComparable<pair<T, U>>
{
    public T v1;
    public U v2;
    public pair(T v1, U v2) {
        this.v1 = v1;
        this.v2 = v2;
    }
    public int CompareTo(pair<T, U> a) {
        int c = Comparer<T>.Default.Compare(v1, a.v1);
        return c != 0 ? c : Comparer<U>.Default.Compare(v2, a.v2);
    }
    public override string ToString() => v1 + " " + v2;
    public void Deconstruct(out T a, out U b) {
        a = v1; b = v2;
    }
}
static class util
{
    public static pair<T, T> make_pair<T>(this IList<T> l) => make_pair(l[0], l[1]);
    public static pair<T, U> make_pair<T, U>(T v1, U v2) => new pair<T, U>(v1, v2);
    public static T sq<T>(T a) => Operator<T>.Multiply(a, a);
    public static T Max<T>(params T[] a) => a.Max();
    public static T Min<T>(params T[] a) => a.Min();
    public static bool inside(int i, int j, int h, int w) => i >= 0 && i < h && j >= 0 && j < w;
    public static void swap<T>(ref T a, ref T b) where T : struct { var t = a; a = b; b = t; }
    public static void swap<T>(this IList<T> a, int i, int j) where T : struct { var t = a[i]; a[i] = a[j]; a[j] = t; }
    public static T[] copy<T>(this IList<T> a) {
        var ret = new T[a.Count];
        for (int i = 0; i < a.Count; i++) ret[i] = a[i];
        return ret;
    }
}
static class Operator<T>
{
    static readonly ParameterExpression x = Expression.Parameter(typeof(T), "x");
    static readonly ParameterExpression y = Expression.Parameter(typeof(T), "y");
    public static readonly Func<T, T, T> Add = Lambda(Expression.Add);
    public static readonly Func<T, T, T> Subtract = Lambda(Expression.Subtract);
    public static readonly Func<T, T, T> Multiply = Lambda(Expression.Multiply);
    public static readonly Func<T, T, T> Divide = Lambda(Expression.Divide);
    public static readonly Func<T, T> Plus = Lambda(Expression.UnaryPlus);
    public static readonly Func<T, T> Negate = Lambda(Expression.Negate);
    public static Func<T, T, T> Lambda(Binary op) => Expression.Lambda<Func<T, T, T>>(op(x, y), x, y).Compile();
    public static Func<T, T> Lambda(Unary op) => Expression.Lambda<Func<T, T>>(op(x), x).Compile();
}

class Scan
{
    public int Int => int.Parse(Str);
    public long Long => long.Parse(Str);
    public double Double => double.Parse(Str);
    public string Str => Console.ReadLine().Trim();
    public pair<T, U> Pair<T, U>() where T : IComparable<T> where U : IComparable<U> {
        T a; U b;
        Multi(out a, out b);
        return new pair<T, U>(a, b);
    }
    public P P {
        get {
            int a, b;
            Multi(out a, out b);
            return new P(a, b);
        }
    }
    public int[] IntArr => StrArr.Select(int.Parse).ToArray();
    public long[] LongArr => StrArr.Select(long.Parse).ToArray();
    public double[] DoubleArr => StrArr.Select(double.Parse).ToArray();
    public string[] StrArr => Str.Split(new[]{' '}, StringSplitOptions.RemoveEmptyEntries);
    bool eq<T, U>() => typeof(T).Equals(typeof(U));
    T ct<T, U>(U a) => (T)Convert.ChangeType(a, typeof(T));
    T cv<T>(string s) => eq<T, int>()    ? ct<T, int>(int.Parse(s))
                       : eq<T, long>()   ? ct<T, long>(long.Parse(s))
                       : eq<T, double>() ? ct<T, double>(double.Parse(s))
                       : eq<T, char>()   ? ct<T, char>(s[0])
                                         : ct<T, string>(s);
    public void Multi<T>(out T a) => a = cv<T>(Str);
    public void Multi<T, U>(out T a, out U b)
    { var ar = StrArr; a = cv<T>(ar[0]); b = cv<U>(ar[1]); }
    public void Multi<T, U, V>(out T a, out U b, out V c)
    { var ar = StrArr; a = cv<T>(ar[0]); b = cv<U>(ar[1]); c = cv<V>(ar[2]); }
    public void Multi<T, U, V, W>(out T a, out U b, out V c, out W d)
    { var ar = StrArr; a = cv<T>(ar[0]); b = cv<U>(ar[1]); c = cv<V>(ar[2]); d = cv<W>(ar[3]); }
    public void Multi<T, U, V, W, X>(out T a, out U b, out V c, out W d, out X e)
    { var ar = StrArr; a = cv<T>(ar[0]); b = cv<U>(ar[1]); c = cv<V>(ar[2]); d = cv<W>(ar[3]); e = cv<X>(ar[4]); }
    public void Multi<T, U, V, W, X, Y>(out T a, out U b, out V c, out W d, out X e, out Y f)
    { var ar = StrArr; a = cv<T>(ar[0]); b = cv<U>(ar[1]); c = cv<V>(ar[2]); d = cv<W>(ar[3]); e = cv<X>(ar[4]); f = cv<Y>(ar[5]); }
}

static class MyMath
{
    public static long Mod = 1000000007;
    public static bool isprime(long a) {
        if (a < 2) return false;
        for (long i = 2; i * i <= a; i++) if (a % i == 0) return false;
        return true;
    }
    public static bool[] sieve(int n) {
        var p = new bool[n + 1];
        for (int i = 2; i <= n; i++) p[i] = true;
        for (int i = 2; i * i <= n; i++) if (p[i]) for (int j = i * i; j <= n; j += i) p[j] = false;
        return p;
    }
    public static List<int> getprimes(int n) {
        var prs = new List<int>();
        var p = sieve(n);
        for (int i = 2; i <= n; i++) if (p[i]) prs.Add(i);
        return prs;
    }
    public static long pow(long a, long b) {
        a %= Mod;
        if (a == 0) return 0;
        if (b == 0) return 1;
        var t = pow(a, b / 2);
        if ((b & 1) == 0) return t * t % Mod;
        return t * t % Mod * a % Mod;
    }
    public static long inv(long a) => pow(a, Mod - 2);
    public static long gcd(long a, long b) {
        while (b > 0) { var t = a % b; a = b; b = t; } return a;
    }
    // a x + b y = gcd(a, b)
    public static long extgcd(long a, long b, out long x, out long y) {
        long g = a; x = 1; y = 0;
        if (b > 0) { g = extgcd(b, a % b, out y, out x); y -= a / b * x; }
        return g;
    }
    public static long lcm(long a, long b) => a / gcd(a, b) * b;

    static long[] facts, invs;
    public static void setfacts(int n) {
        facts = new long[n + 1];
        facts[0] = 1;
        for (int i = 1; i <= n; i++) facts[i] = facts[i - 1] * i % Mod;
        invs = new long[n + 1];
        invs[n] = inv(facts[n]);
        for (int i = n; i > 0 ; i--) invs[i - 1] = invs[i] * i % Mod;
    }
    public static long comb(long n, long r) {
        if (n < 0 || r < 0 || r > n) return 0;
        if (facts != null && facts.Length > n) return facts[n] * invs[r] % Mod * invs[n - r] % Mod;
        if (n - r < r) r = n - r;
        long numer = 1, denom = 1;
        for (long i = 0; i < r; i++) {
            numer = numer * ((n - i) % Mod) % Mod;
            denom = denom * ((i + 1) % Mod) % Mod;
        }
        return numer * inv(denom) % Mod;
    }
    public static long[][] getcombs(int n) {
        var ret = new long[n + 1][];
        for (int i = 0; i <= n; i++) {
            ret[i] = new long[i + 1];
            ret[i][0] = ret[i][i] = 1;
            for (int j = 1; j < i; j++) ret[i][j] = (ret[i - 1][j - 1] + ret[i - 1][j]) % Mod;
        }
        return ret;
    }
    // nC0, nC2, ..., nCn
    public static long[] getcomb(int n) {
        var ret = new long[n + 1];
        ret[0] = 1;
        for (int i = 0; i < n; i++) ret[i + 1] = ret[i] * (n - i) % Mod * inv(i + 1) % Mod;
        return ret;
    }
    public static bool nextPermutation<T>(IList<T> p) where T : struct, IComparable<T> {
        for (int i = p.Count - 2; i >= 0; --i) {
            if (p[i].CompareTo(p[i + 1]) < 0) {
                for (int j = p.Count - 1; ; --j) {
                    if (p[j].CompareTo(p[i]) > 0) {
                        p.swap(i, j);
                        for(++i, j = p.Count - 1; i < j; ++i, --j)
                            p.swap(i, j);

                        return true;
                    }
                }
            }
        }
        return false;
    }
    public static bool nextPermutation<T>(IList<T> p, Comparison<T> compare) where T : struct {
        for (int i = p.Count - 2; i >= 0; --i) {
            if (compare(p[i], p[i + 1]) < 0) {
                for (int j = p.Count - 1; ; --j) {
                    if (compare(p[j], p[i]) > 0) {
                        p.swap(i, j);
                        for (++i, j = p.Count - 1; i < j; ++i, --j)
                            p.swap(i, j);

                        return true;
                    }
                }
            }
        }
        return false;
    }
}