Problem:
Score : 1300 points

Problem Statement

We have a graph with N vertices, numbered 0 through N-1. Edges are yet to be added.

We will process Q queries to add edges.
In the i-th (1≦i≦Q) query, three integers A_i, B_i and C_i will be given, and we will add infinitely many edges to the graph as follows:

The two vertices numbered A_i and B_i will be connected by an edge with a weight of C_i.

The two vertices numbered B_i and A_i+1 will be connected by an edge with a weight of C_i+1.

The two vertices numbered A_i+1 and B_i+1 will be connected by an edge with a weight of C_i+2.

The two vertices numbered B_i+1 and A_i+2 will be connected by an edge with a weight of C_i+3.

The two vertices numbered A_i+2 and B_i+2 will be connected by an edge with a weight of C_i+4.

The two vertices numbered B_i+2 and A_i+3 will be connected by an edge with a weight of C_i+5.

The two vertices numbered A_i+3 and B_i+3 will be connected by an edge with a weight of C_i+6.

...

Here, consider the indices of the vertices modulo N.
For example, the vertice numbered N is the one numbered 0, and the vertice numbered 2N-1 is the one numbered N-1.

The figure below shows the first seven edges added when N=16, A_i=7, B_i=14, C_i=1:

After processing all the queries, find the total weight of the edges contained in a minimum spanning tree of the graph.

Constraints

2≦N≦200,000

1≦Q≦200,000

0≦A_i,B_i≦N-1

1≦C_i≦10^9

Input

The input is given from Standard Input in the following format:

N Q
A_1 B_1 C_1
A_2 B_2 C_2
:
A_Q B_Q C_Q

Output

Print the total weight of the edges contained in a minimum spanning tree of the graph.

Sample Input 1

7 1
5 2 1

Sample Output 1

21

The figure below shows the minimum spanning tree of the graph:

Note that there can be multiple edges connecting the same pair of vertices.

Sample Input 2

2 1
0 0 1000000000

Sample Output 2

1000000001

Also note that there can be self-loops.

Sample Input 3

5 3
0 1 10
0 2 10
0 4 10

Sample Output 3

42

Solution in C# (Mono 4.6.2.0) (judge verdict: Time Limit Exceeded)

using System;
using System.Linq;
using System.Linq.Expressions;
using System.Collections.Generic;
using Debug = System.Diagnostics.Debug;
using StringBuilder = System.Text.StringBuilder;
using System.Numerics;
using Point = System.Numerics.Complex;
using Number = System.Int64;
using C = System.Int32;
using V = System.Int32;
namespace Program
{
    public class Solver
    {
        public void Solve()
        {
            var n = sc.Integer();
            var q = sc.Integer();
            var pq = new BinaryHeapPriorityQueue<Triplet<int, int, long>>((l, r) => l.K.CompareTo(r.K));
            var s = new DisjointSet(n);
            for (int i = 0; i < q; i++)
            {
                var a = sc.Integer() % n;
                var b = sc.Integer() % n;
                var c = sc.Long();
                if (a != b)
                    pq.Enqueue(new Triplet<C, C, Number>(a, b, c));
                if ((a + 1) % n != b)
                    pq.Enqueue(new Triplet<C, C, Number>((a + 1) % n, b, c + 1));
            }
            var ans = 0L;
            var cnt = 0;
            var E = new HashMap<long>();
            while (cnt < n - 1)
            {
                var p = pq.Dequeue();
                Debug.WriteLine(p);
                if (s.Unite(p.I, p.J))
                {
                    //Debug.WriteLine("{0} {1} {2}", p.I, p.J, p.K);
                    ans += p.K;
                    cnt++;
                }

                {
                    var u = p.I;
                    var v = p.J;
                    if (u > v) Swap(ref u, ref v);
                    var id = u * 200000 + v;
                    if (E[id] <= p.K + 2) continue;
                }
                pq.Enqueue(new Triplet<C, C, Number>((p.I + 1) % n, (p.J + 1) % n, p.K + 2));
            }
            IO.Printer.Out.WriteLine(ans);



        }

        public IO.StreamScanner sc = new IO.StreamScanner(Console.OpenStandardInput());
        static T[] Enumerate<T>(int n, Func<int, T> f) { var a = new T[n]; for (int i = 0; i < n; ++i) a[i] = f(i); return a; }
        static public void Swap<T>(ref T a, ref T b) { var tmp = a; a = b; b = tmp; }
    }
}
#region main
static class Ex
{
    static public string AsString(this IEnumerable<char> ie) { return new string(System.Linq.Enumerable.ToArray(ie)); }
    static public string AsJoinedString<T>(this IEnumerable<T> ie, string st = " ") { return string.Join(st, ie); }
    static public void Main()
    {
        var solver = new Program.Solver();
        solver.Solve();
        Program.IO.Printer.Out.Flush();
    }
}
#endregion
#region Ex
namespace Program.IO
{
    using System.IO;
    using System.Text;
    using System.Globalization;
    public class Printer: StreamWriter
    {
        static Printer() { Out = new Printer(Console.OpenStandardOutput()) { AutoFlush = false }; }
        public static Printer Out { get; set; }
        public override IFormatProvider FormatProvider { get { return CultureInfo.InvariantCulture; } }
        public Printer(System.IO.Stream stream) : base(stream, new UTF8Encoding(false, true)) { }
        public Printer(System.IO.Stream stream, Encoding encoding) : base(stream, encoding) { }
        public void Write<T>(string format, T[] source) { base.Write(format, source.OfType<object>().ToArray()); }
        public void WriteLine<T>(string format, T[] source) { base.WriteLine(format, source.OfType<object>().ToArray()); }
    }
    public class StreamScanner
    {
        public StreamScanner(Stream stream) { str = stream; }
        public readonly Stream str;
        private readonly byte[] buf = new byte[1024];
        private int len, ptr;
        public bool isEof = false;
        public bool IsEndOfStream { get { return isEof; } }
        private byte read()
        {
            if (isEof) return 0;
            if (ptr >= len) { ptr = 0; if ((len = str.Read(buf, 0, 1024)) <= 0) { isEof = true; return 0; } }
            return buf[ptr++];
        }
        public char Char() { byte b = 0; do b = read(); while ((b < 33 || 126 < b) && !isEof); return (char)b; }

        public string Scan()
        {
            var sb = new StringBuilder();
            for (var b = Char(); b >= 33 && b <= 126; b = (char)read())
                sb.Append(b);
            return sb.ToString();
        }
        public string ScanLine()
        {
            var sb = new StringBuilder();
            for (var b = Char(); b != '\n'; b = (char)read())
                if (b == 0) break;
                else if (b != '\r') sb.Append(b);
            return sb.ToString();
        }
        public long Long()
        {
            if (isEof) return long.MinValue;
            long ret = 0; byte b = 0; var ng = false;
            do b = read();
            while (b != 0 && b != '-' && (b < '0' || '9' < b));
            if (b == 0) return long.MinValue;
            if (b == '-') { ng = true; b = read(); }
            for (; true; b = read())
            {
                if (b < '0' || '9' < b)
                    return ng ? -ret : ret;
                else ret = ret * 10 + b - '0';
            }
        }
        public int Integer() { return (isEof) ? int.MinValue : (int)Long(); }
        public double Double() { var s = Scan(); return s != "" ? double.Parse(s, CultureInfo.InvariantCulture) : double.NaN; }
        private T[] enumerate<T>(int n, Func<T> f)
        {
            var a = new T[n];
            for (int i = 0; i < n; ++i) a[i] = f();
            return a;
        }

        public char[] Char(int n) { return enumerate(n, Char); }
        public string[] Scan(int n) { return enumerate(n, Scan); }
        public double[] Double(int n) { return enumerate(n, Double); }
        public int[] Integer(int n) { return enumerate(n, Integer); }
        public long[] Long(int n) { return enumerate(n, Long); }
    }
}
#endregion

#region HashMap<K,V>
public class HashMap<K>: Dictionary<K, long>
//where V : new()
{
    public HashMap() : base() { }
    public HashMap(int cap) : base(cap) { }
    new public long this[K i]
    {
        get
        {
            long v;
            return TryGetValue(i, out v) ? v :
                base[i] = 1L << 60;
            //base[i] = new V();
        }
        set { base[i] = value; }
    }
}
#endregion

#region PriorityQueue by BinaryHeap
public class BinaryHeapPriorityQueue<T>
{
    private readonly List<T> heap;
    private readonly Comparison<T> compare;
    private int size;
    public BinaryHeapPriorityQueue() : this(Comparer<T>.Default) { }
    public BinaryHeapPriorityQueue(IComparer<T> comparer) : this(16, comparer.Compare) { }
    public BinaryHeapPriorityQueue(Comparison<T> comparison) : this(16, comparison) { }
    public BinaryHeapPriorityQueue(int capacity, Comparison<T> comparison)
    {
        heap = new List<T>(capacity);
        compare = comparison;
    }
    public void Enqueue(T item)
    {

        heap.Add(item);
        var i = size++;
        while (i > 0)
        {
            var p = (i - 1) >> 1;
            if (compare(heap[p], item) <= 0)
                break;
            heap[i] = heap[p];
            i = p;
        }
        heap[i] = item;

    }
    public T Dequeue()
    {
        var ret = heap[0];
        var x = heap[--size];
        var i = 0;
        while ((i << 1) + 1 < size)
        {
            var a = (i << 1) + 1;
            var b = (i << 1) + 2;
            if (b < size && compare(heap[b], heap[a]) < 0) a = b;
            if (compare(heap[a], x) >= 0)
                break;
            heap[i] = heap[a];
            i = a;
        }
        heap[i] = x;
        heap.RemoveAt(size);
        return ret;

    }
    public T Peek() { return heap[0]; }
    public int Count { get { return size; } }
    public bool Any() { return size > 0; }
}
#endregion
#region Triplet<ST,FT,TT>
static public class Triplet
{
    static public Triplet<FT, ST, TT> Create<FT, ST, TT>(FT f, ST s, TT t) where FT : IComparable<FT> where ST : IComparable<ST> where TT : IComparable<TT>
    {
        return new Triplet<FT, ST, TT>(f, s, t);
    }

    static public Triplet<FT, ST, TT> Max<FT, ST, TT>(Triplet<FT, ST, TT> p, Triplet<FT, ST, TT> q) where FT : IComparable<FT> where ST : IComparable<ST> where TT : IComparable<TT>
    {
        return p.CompareTo(q) >= 0 ? p : q;
    }
    static public Triplet<FT, ST, TT> Min<FT, ST, TT>(Triplet<FT, ST, TT> p, Triplet<FT, ST, TT> q) where FT : IComparable<FT> where ST : IComparable<ST> where TT : IComparable<TT>
    {
        return p.CompareTo(q) <= 0 ? p : q;
    }
}
public struct Triplet<FT, ST, TT>: IComparable<Triplet<FT, ST, TT>> where FT : IComparable<FT> where ST : IComparable<ST> where TT : IComparable<TT>
{
    public FT I;
    public ST J;
    public TT K;
    public Triplet(FT i, ST j, TT k) : this() { I = i; J = j; K = k; }
    public int CompareTo(Triplet<FT, ST, TT> other)
    {
        var cmp = I.CompareTo(other.I);
        if (cmp != 0) return cmp;
        cmp = J.CompareTo(other.J);
        if (cmp != 0) return cmp;
        return K.CompareTo(other.K);
    }
    public override string ToString() { return string.Format("{0} {1} {2}", I, J, K); }
}
#endregion
#region DisjointSet
public class DisjointSet
{
    int[] par;
    byte[] rank;
    public DisjointSet(int n)
    {
        par = new int[n];
        for (int i = 0; i < n; i++)
            par[i] = -1;
        rank = new byte[n];
    }
    public int this[int id]
    {
        get
        {
            if ((par[id] < 0)) return id;
            return par[id] = this[par[id]];
        }
    }
    public bool Unite(int x, int y)
    {
        x = this[x]; y = this[y];
        if (x == y) return false;
        if (rank[x] < rank[y])
        {
            par[y] += par[x];
            par[x] = y;
        }
        else
        {
            par[x] += par[y];
            par[y] = x;
            if (rank[x] == rank[y])
                rank[x]++;
        }
        return true;
    }
    public int Size(int x) { return -par[this[x]]; }
    public bool IsUnited(int x, int y) { return this[x] == this[y]; }

}
#endregion