Problem:
Score: 600 points

Problem Statement

M-kun is a brilliant air traffic controller.

On the display of his radar, there are N airplanes numbered 1, 2, ..., N, all flying at the same altitude.

Each of the airplanes flies at a constant speed of 0.1 per second in a constant direction. The current coordinates of the airplane numbered i are (X_i, Y_i), and the direction of the airplane is as follows:

if U_i is U, it flies in the positive y direction;

if U_i is R, it flies in the positive x direction;

if U_i is D, it flies in the negative y direction;

if U_i is L, it flies in the negative x direction.

To help M-kun in his work, determine whether there is a pair of airplanes that will collide with each other if they keep flying as they are now.

If there is such a pair, find the number of seconds after which the first collision will happen.

We assume that the airplanes are negligibly small so that two airplanes only collide when they reach the same coordinates simultaneously.

Constraints

1 \leq N \leq 200000

0 \leq X_i, Y_i \leq 200000

U_i is U, R, D, or L.

The current positions of the N airplanes, (X_i, Y_i), are all distinct.

N, X_i, and Y_i are integers.

Input

Input is given from Standard Input in the following format:

N
X_1 Y_1 U_1
X_2 Y_2 U_2
X_3 Y_3 U_3
:
X_N Y_N U_N

Output

If there is a pair of airplanes that will collide with each other if they keep flying as they are now, print an integer representing the number of seconds after which the first collision will happen.

If there is no such pair, print SAFE.

Sample Input 1

2
11 1 U
11 47 D

Sample Output 1

230

If the airplanes keep flying as they are now, two airplanes numbered 1 and 2 will reach the coordinates (11, 24) simultaneously and collide.

Sample Input 2

4
20 30 U
30 20 R
20 10 D
10 20 L

Sample Output 2

SAFE

No pair of airplanes will collide.

Sample Input 3

8
168 224 U
130 175 R
111 198 D
121 188 L
201 116 U
112 121 R
145 239 D
185 107 L

Sample Output 3

100

Solution in C# (.NET Core 3.1.201) (judge verdict: Wrong Answer)

using System;
using System.Collections.Generic;
using System.Linq;
using System.IO;
using System.Threading;
using System.Runtime.CompilerServices;
using System.Text;
using System.Diagnostics;
using System.Numerics;
using static System.Console;
using static System.Convert;
using static System.Math;
using static Template;
using SC = Scanner;

public partial class Solver
{
    public void Solve()
    {
        var N = sc.Int;
        var P = new (int X, int Y, char U)[N];

        for (int i = 0; i < N; i++)
        {
            sc.Make(out P[i].X, out P[i].Y, out P[i].U);

        }
        var res = int.MaxValue;
        Array.Sort(P, (a, b) => { if (a.X == b.X) return a.Y - b.Y; return a.X - b.X; });
        const int g = 200200;
        var umin = Create(400400, () => int.MaxValue / 2);
        var dmin = Create(400400, () => int.MaxValue / 2);
        for (int i = 0; i < N;)
        {
            int idx = i;
            var U = new Set<int>((a, b) => b - a);
            var D = new Set<int>();
            while (idx < N && P[idx].X == P[i].X)
            {
                if (P[idx].U == 'U')
                {
                    var lw = D.LowerBound(P[idx].Y);
                    if (lw != D.Count) chmin(ref res, (D[lw] - P[idx].Y) * 5);
                    U.Add(P[idx].Y);
                    umin[P[idx].Y - P[idx].X + g] = P[idx].X;
                }
                else if (P[idx].U == 'D')
                {
                    var lw = U.LowerBound(P[idx].Y);
                    if (lw != U.Count) chmin(ref res, (P[idx].Y - U[lw]) * 5);
                    D.Add(P[idx].Y);
                    dmin[P[idx].Y + P[idx].X] = P[idx].X;
                }
                else if (P[idx].U == 'L')
                {
                    if (umin[P[i].Y - P[i].X + g] != int.MaxValue / 2)
                        chmin(ref res, -(umin[P[idx].Y - P[idx].X + g] - P[idx].X) * 10);
                    if (dmin[P[i].Y + P[i].X] != int.MaxValue / 2)
                        chmin(ref res, -(dmin[P[idx].Y + P[idx].X] - P[idx].X) * 10);
                }
                idx++;
            }
            i = idx;
        }
        for (int i = 0; i < dmin.Length; i++) umin[i] = dmin[i] = int.MaxValue / 2;
        for (int i = N - 1; i >= 0; i--)
        {
            if (P[i].U == 'U')
                umin[P[i].Y + P[i].X] = P[i].X;
            else if (P[i].U == 'D')
                dmin[P[i].Y - P[i].X + g] = P[i].X;
            else if (P[i].U == 'R')
            {
                if (umin[P[i].Y + P[i].X] != int.MaxValue / 2)
                    chmin(ref res, -(P[i].X - umin[P[i].Y + P[i].X]) * 10);
                if (dmin[P[i].Y - P[i].X + g] != int.MaxValue / 2)
                    chmin(ref res, -(P[i].X - dmin[P[i].Y - P[i].X + g]) * 10);
            }
        }
        Array.Sort(P, (a, b) => { if (a.Y == b.Y) return a.X - b.X; return a.Y - b.Y; });
        for (int i = 0; i < N;)
        {
            int idx = i;
            var R = new Set<int>((a, b) => b - a);
            var L = new Set<int>();
            while (idx < N && P[idx].Y == P[i].Y)
            {
                if (P[idx].U == 'R')
                {
                    var lw = L.LowerBound(P[idx].X);
                    if (lw != L.Count) chmin(ref res, (L[lw] - P[idx].X) * 5);
                    R.Add(P[idx].Y);
                }
                else if (P[idx].U == 'L')
                {
                    var lw = R.LowerBound(P[idx].X);
                    if (lw != R.Count) chmin(ref res, (P[idx].X - R[lw]) * 5);
                    L.Add(P[idx].Y);
                }
                idx++;
            }
            i = idx;
        }
        if (res == int.MaxValue) Console.WriteLine("SAFE");
        else Console.WriteLine(res);
    }
}

public class Set<T>
{
    Node root;
    readonly IComparer<T> comparer;
    readonly Node nil;
    public bool IsMultiSet { get; set; }
    public Set(IComparer<T> comparer)
    {
        nil = new Node(default(T));
        root = nil;
        this.comparer = comparer;
    }
    public Set(Comparison<T> comaprison) : this(Comparer<T>.Create(comaprison)) { }
    public Set() : this(Comparer<T>.Default) { }
    public bool Add(T v)
    {
        return insert(ref root, v);
    }
    public bool Remove(T v)
    {
        return remove(ref root, v);
    }
    public T this[int index] { get { return find(root, index); } }
    public int Count { get { return root.Count; } }
    public void RemoveAt(int k)
    {
        if (k < 0 || k >= root.Count) throw new ArgumentOutOfRangeException();
        removeAt(ref root, k);
    }
    public T[] Items
    {
        get
        {
            var ret = new T[root.Count];
            var k = 0;
            walk(root, ret, ref k);
            return ret;
        }
    }
    void walk(Node t, T[] a, ref int k)
    {
        if (t.Count == 0) return;
        walk(t.lst, a, ref k);
        a[k++] = t.Key;
        walk(t.rst, a, ref k);
    }

    bool insert(ref Node t, T key)
    {
        if (t.Count == 0) { t = new Node(key); t.lst = t.rst = nil; t.Update(); return true; }
        var cmp = comparer.Compare(t.Key, key);
        bool res;
        if (cmp > 0)
            res = insert(ref t.lst, key);
        else if (cmp == 0)
        {
            if (IsMultiSet) res = insert(ref t.lst, key);
            else return false;
        }
        else res = insert(ref t.rst, key);
        balance(ref t);
        return res;
    }
    bool remove(ref Node t, T key)
    {
        if (t.Count == 0) return false;
        var cmp = comparer.Compare(key, t.Key);
        bool ret;
        if (cmp < 0) ret = remove(ref t.lst, key);
        else if (cmp > 0) ret = remove(ref t.rst, key);
        else
        {
            ret = true;
            var k = t.lst.Count;
            if (k == 0) { t = t.rst; return true; }
            if (t.rst.Count == 0) { t = t.lst; return true; }


            t.Key = find(t.lst, k - 1);
            removeAt(ref t.lst, k - 1);
        }
        balance(ref t);
        return ret;
    }
    void removeAt(ref Node t, int k)
    {
        var cnt = t.lst.Count;
        if (cnt < k) removeAt(ref t.rst, k - cnt - 1);
        else if (cnt > k) removeAt(ref t.lst, k);
        else
        {
            if (cnt == 0) { t = t.rst; return; }
            if (t.rst.Count == 0) { t = t.lst; return; }

            t.Key = find(t.lst, k - 1);
            removeAt(ref t.lst, k - 1);
        }
        balance(ref t);
    }
    void balance(ref Node t)
    {
        var balance = t.lst.Height - t.rst.Height;
        if (balance == -2)
        {
            if (t.rst.lst.Height - t.rst.rst.Height > 0) { rotR(ref t.rst); }
            rotL(ref t);
        }
        else if (balance == 2)
        {
            if (t.lst.lst.Height - t.lst.rst.Height < 0) rotL(ref t.lst);
            rotR(ref t);
        }
        else t.Update();
    }

    T find(Node t, int k)
    {
        if (k < 0 || k > root.Count) throw new ArgumentOutOfRangeException();
        for (; ; )
        {
            if (k == t.lst.Count) return t.Key;
            else if (k < t.lst.Count) t = t.lst;
            else { k -= t.lst.Count + 1; t = t.rst; }
        }
    }
    public int LowerBound(T v)
    {
        var k = 0;
        var t = root;
        for (; ; )
        {
            if (t.Count == 0) return k;
            if (comparer.Compare(v, t.Key) <= 0) t = t.lst;
            else { k += t.lst.Count + 1; t = t.rst; }
        }
    }
    public int UpperBound(T v)
    {
        var k = 0;
        var t = root;
        for (; ; )
        {
            if (t.Count == 0) return k;
            if (comparer.Compare(t.Key, v) <= 0) { k += t.lst.Count + 1; t = t.rst; }
            else t = t.lst;
        }
    }
    public bool Contains(T v)
    {
        var k = 0;
        var t = root;
        for (; ; )
        {
            if (t.Count == 0) return false;
            if (comparer.Compare(t.Key, v) == 0) return true;
            if (comparer.Compare(t.Key, v) < 0) { k += t.lst.Count + 1; t = t.rst; }
            else t = t.lst;
        }
    }
    void rotR(ref Node t)
    {
        var l = t.lst;
        t.lst = l.rst;
        l.rst = t;
        t.Update();
        l.Update();
        t = l;
    }
    void rotL(ref Node t)
    {
        var r = t.rst;
        t.rst = r.lst;
        r.lst = t;
        t.Update();
        r.Update();
        t = r;
    }
    public T[] ToArray()
    {
        var rt = new T[Count]; var idx = 0;
        Action<Node> dfs = null;
        dfs = a =>
        {
            if (a.lst != nil) dfs(a.lst);
            rt[idx++] = a.Key;
            if (a.rst != nil) dfs(a.rst);
        };
        dfs(root);
        return rt;
    }

    class Node
    {
        public Node(T key)
        {
            Key = key;
        }
        public int Count { get; private set; }
        public sbyte Height { get; private set; }
        public T Key { get; set; }
        public Node lst, rst;
        public void Update()
        {
            Count = 1 + lst.Count + rst.Count;
            Height = (sbyte)(1 + Math.Max(lst.Height, rst.Height));
        }
        public override string ToString()
        {
            return string.Format("Count = {0}, Key = {1}", Count, Key);
        }
    }
}
#region Template
public partial class Solver
{
    public SC sc = new SC();
    static void Main(string[] args)
    {
        Console.SetOut(new StreamWriter(Console.OpenStandardOutput()) { AutoFlush = false });
        var sol = new Solver();
        int testcase = 1;
        //testcase = sol.sc.Int;
        //var th = new Thread(sol.Solve, 1 << 26);th.Start();th.Join();
        while (testcase-- > 0)
            sol.Solve();
        Console.Out.Flush();
    }
}
public static class Template
{
    [MethodImpl(MethodImplOptions.AggressiveInlining)]
    public static bool chmin<T>(ref T a, T b) where T : IComparable<T> { if (a.CompareTo(b) > 0) { a = b; return true; } return false; }
    [MethodImpl(MethodImplOptions.AggressiveInlining)]
    public static bool chmax<T>(ref T a, T b) where T : IComparable<T> { if (a.CompareTo(b) < 0) { a = b; return true; } return false; }
    [MethodImpl(MethodImplOptions.AggressiveInlining)]
    public static void swap<T>(ref T a, ref T b) { var t = b; b = a; a = t; }
    [MethodImpl(MethodImplOptions.AggressiveInlining)]
    public static void swap<T>(this IList<T> A, int i, int j) { var t = A[i]; A[i] = A[j]; A[j] = t; }
    public static T[] Create<T>(int n, Func<T> f) { var rt = new T[n]; for (var i = 0; i < rt.Length; ++i) rt[i] = f(); return rt; }
    public static T[] Create<T>(int n, Func<int, T> f) { var rt = new T[n]; for (var i = 0; i < rt.Length; ++i) rt[i] = f(i); return rt; }
    public static void Out<T>(this IList<T> A, out T a) => a = A[0];
    public static void Out<T>(this IList<T> A, out T a, out T b) { a = A[0]; b = A[1]; }
    public static void Out<T>(this IList<T> A, out T a, out T b, out T c) { A.Out(out a, out b); c = A[2]; }
    public static void Out<T>(this IList<T> A, out T a, out T b, out T c, out T d) { A.Out(out a, out b, out c); d = A[3]; }
    public static string Concat<T>(this IEnumerable<T> A, string sp) => string.Join(sp, A);
    public static char ToChar(this int s, char begin = '0') => (char)(s + begin);
    public static IEnumerable<T> Shuffle<T>(this IEnumerable<T> A) => A.OrderBy(v => Guid.NewGuid());
    public static int CompareTo<T>(this T[] A, T[] B, Comparison<T> cmp = null) { cmp = cmp ?? Comparer<T>.Default.Compare; for (var i = 0; i < Min(A.Length, B.Length); i++) { int c = cmp(A[i], B[i]); if (c > 0) return 1; else if (c < 0) return -1; } if (A.Length == B.Length) return 0; if (A.Length > B.Length) return 1; else return -1; }
    public static string ToStr<T>(this T[][] A) => A.Select(a => a.Concat(" ")).Concat("\n");
    public static int ArgMax<T>(this IList<T> A, Comparison<T> cmp = null) { cmp = cmp ?? Comparer<T>.Default.Compare; T max = A[0]; int rt = 0; for (int i = 1; i < A.Count; i++) if (cmp(max, A[i]) < 0) { max = A[i]; rt = i; } return rt; }
    public static T PopBack<T>(this List<T> A) { var v = A[A.Count - 1]; A.RemoveAt(A.Count - 1); return v; }
    public static void Fail<T>(T s) { Console.WriteLine(s); Console.Out.Close(); Environment.Exit(0); }
}
public class Scanner
{
    public string Str => Console.ReadLine().Trim();
    public int Int => int.Parse(Str);
    public long Long => long.Parse(Str);
    public double Double => double.Parse(Str);
    public int[] ArrInt => Str.Split(' ').Select(int.Parse).ToArray();
    public long[] ArrLong => Str.Split(' ').Select(long.Parse).ToArray();
    public char[][] Grid(int n) => Create(n, () => Str.ToCharArray());
    public int[] ArrInt1D(int n) => Create(n, () => Int);
    public long[] ArrLong1D(int n) => Create(n, () => Long);
    public int[][] ArrInt2D(int n) => Create(n, () => ArrInt);
    public long[][] ArrLong2D(int n) => Create(n, () => ArrLong);
    private Queue<string> q = new Queue<string>();
    [MethodImpl(MethodImplOptions.AggressiveInlining)]
    public T Next<T>() { if (q.Count == 0) foreach (var item in Str.Split(' ')) q.Enqueue(item); return (T)Convert.ChangeType(q.Dequeue(), typeof(T)); }
    public void Make<T1>(out T1 v1) => v1 = Next<T1>();
    public void Make<T1, T2>(out T1 v1, out T2 v2) { v1 = Next<T1>(); v2 = Next<T2>(); }
    public void Make<T1, T2, T3>(out T1 v1, out T2 v2, out T3 v3) { Make(out v1, out v2); v3 = Next<T3>(); }
    public void Make<T1, T2, T3, T4>(out T1 v1, out T2 v2, out T3 v3, out T4 v4) { Make(out v1, out v2, out v3); v4 = Next<T4>(); }
    public void Make<T1, T2, T3, T4, T5>(out T1 v1, out T2 v2, out T3 v3, out T4 v4, out T5 v5) { Make(out v1, out v2, out v3, out v4); v5 = Next<T5>(); }
    public void Make<T1, T2, T3, T4, T5, T6>(out T1 v1, out T2 v2, out T3 v3, out T4 v4, out T5 v5, out T6 v6) { Make(out v1, out v2, out v3, out v4, out v5); v6 = Next<T6>(); }
    public void Make<T1, T2, T3, T4, T5, T6, T7>(out T1 v1, out T2 v2, out T3 v3, out T4 v4, out T5 v5, out T6 v6, out T7 v7) { Make(out v1, out v2, out v3, out v4, out v5, out v6); v7 = Next<T7>(); }
    public (T1, T2) Make<T1, T2>() { Make(out T1 v1, out T2 v2); return (v1, v2); }
    public (T1, T2, T3) Make<T1, T2, T3>() { Make(out T1 v1, out T2 v2, out T3 v3); return (v1, v2, v3); }
    public (T1, T2, T3, T4) Make<T1, T2, T3, T4>() { Make(out T1 v1, out T2 v2, out T3 v3, out T4 v4); return (v1, v2, v3, v4); }
}

#endregion