Problem:
Score : 600 points

Problem Statement

Snuke is introducing a robot arm with the following properties to his factory:

The robot arm consists of m sections and m+1 joints. The sections are numbered 1, 2, ..., m, and the joints are numbered 0, 1, ..., m. Section i connects Joint i-1 and Joint i. The length of Section i is d_i.

For each section, its mode can be specified individually. There are four modes: L, R, D and U. The mode of a section decides the direction of that section. If we consider the factory as a coordinate plane, the position of Joint i will be determined as follows (we denote its coordinates as (x_i, y_i)):

(x_0, y_0) = (0, 0).

If the mode of Section i is L, (x_{i}, y_{i}) = (x_{i-1} - d_{i}, y_{i-1}).

If the mode of Section i is R, (x_{i}, y_{i}) = (x_{i-1} + d_{i}, y_{i-1}).

If the mode of Section i is D, (x_{i}, y_{i}) = (x_{i-1}, y_{i-1} - d_{i}).

If the mode of Section i is U, (x_{i}, y_{i}) = (x_{i-1}, y_{i-1} + d_{i}).

Snuke would like to introduce a robot arm so that the position of Joint m can be matched with all of the N points (X_1, Y_1), (X_2, Y_2), ..., (X_N, Y_N) by properly specifying the modes of the sections.
Is this possible?
If so, find such a robot arm and how to bring Joint m to each point (X_j, Y_j).

Constraints

All values in input are integers.

1 \leq N \leq 1000

-10^9 \leq X_i \leq 10^9

-10^9 \leq Y_i \leq 10^9

Partial Score

In the test cases worth 300 points, -10 \leq X_i \leq 10 and -10 \leq Y_i \leq 10 hold.

Input

Input is given from Standard Input in the following format:

N
X_1 Y_1
X_2 Y_2
:
X_N Y_N

Output

If the condition can be satisfied, follow the following format. If the condition cannot be satisfied, print -1.

m
d_1 d_2 ... d_m
w_1
w_2
:
w_N

m and d_i are the configurations of the robot arm. Refer to the problem statement for what each of them means.
Here, 1 \leq m \leq 40 and 1 \leq d_i \leq 10^{12} must hold. Also, m and d_i must all be integers.

w_j is a string of length m that represents the way to bring Joint m of the robot arm to point (X_j, Y_j).
The i-th character of w_j should be one of the letters L, R, D and U, representing the mode of Section i.

Sample Input 1

3
-1 0
0 3
2 -1

Sample Output 1

2
1 2
RL
UU
DR

In the given way to bring Joint m of the robot arm to each (X_j, Y_j), the positions of the joints will be as follows:

To (X_1, Y_1) = (-1, 0): First, the position of Joint 0 is (x_0, y_0) = (0, 0). As the mode of Section 1 is R, the position of Joint 1 is (x_1, y_1) = (1, 0). Then, as the mode of Section 2 is L, the position of Joint 2 is (x_2, y_2) = (-1, 0).

To (X_2, Y_2) = (0, 3): (x_0, y_0) = (0, 0), (x_1, y_1) = (0, 1), (x_2, y_2) = (0, 3).

To (X_3, Y_3) = (2, -1): (x_0, y_0) = (0, 0), (x_1, y_1) = (0, -1), (x_2, y_2) = (2, -1).

Sample Input 2

5
0 0
1 0
2 0
3 0
4 0

Sample Output 2

-1

Sample Input 3

2
1 1
1 1

Sample Output 3

2
1 1
RU
UR

There may be duplicated points among (X_j, Y_j).

Sample Input 4

3
-7 -3
7 3
-3 -7

Sample Output 4

5
3 1 4 1 5
LRDUL
RDULR
DULRD

Solution in C# (Mono 4.6.2.0) (judge verdict: Wrong Answer)

using System;
using System.Collections.Generic;
using System.IO;
using System.Linq;
using System.Runtime.CompilerServices;
using static System.Math;
using System.Text;

namespace Program
{
    public static class ABC111D
    {
        static public void Solve()
        {
            var N = NextLong;
            var xy = Repeat(0, N).Select(_ => new { x = NextLong, y = NextLong }).ToList();
            var sampleOdd = xy.Select(e => Abs(e.x) + Abs(e.y)).First() % 2;
            if (xy.Any(e => (Abs(e.x) + Abs(e.y)) % 2 != sampleOdd))
            {
                Console.WriteLine(-1);
                return;
            }
            var udeList = new List<long>();
            var maxX = Abs(xy.Select(e => (e.x)).Max());
            var maxY = Abs(xy.Select(e => (e.y)).Max());
            var tmpX = 1L;
            var tmpMX = maxX;
            var tmpSum = 0L;
            while (tmpMX != 0)
            {
                if ((tmpMX & 1) == 1)
                {
                    udeList.Add(tmpX);
                    tmpSum += tmpX;
                }
                tmpX *= 2;
                tmpMX >>= 1;
            }
            var tmpY = 1L;
            var tmpMY = maxY;
            while (tmpMY != 0)
            {
                if ((tmpMY & 1) == 1)
                {
                    udeList.Add(tmpY);
                    tmpSum += tmpY;
                }
                tmpY *= 2;
                tmpMY >>= 1;
            }
            if (tmpSum % 2L != sampleOdd)
            {
                udeList.Add(1);
            }
            udeList = udeList.OrderByDescending(e => e).ToList();
            Console.WriteLine(udeList.Count);
            Console.Write(udeList.First());
            foreach (var item in udeList.Skip(1))
            {
                Console.Write(" ");
                Console.Write(item);
            }
            Console.WriteLine();

            foreach (var item in xy)
            {
                var pointX = 0L;
                var pointY = 0L;
                foreach (var ude in udeList)
                {
                    if (Abs(pointX - item.x) < Abs(pointY - item.y))
                    {
                        if (item.y < pointY)
                        {
                            Console.Write('D');
                            pointY -= ude;
                        }
                        else
                        {
                            Console.Write('U');
                            pointY += ude;
                        }
                    }
                    else
                    {
                        if (item.x < pointX)
                        {
                            Console.Write('L');
                            pointX -= ude;
                        }
                        else
                        {
                            Console.Write('R');
                            pointX += ude;
                        }
                    }
                }
                Console.WriteLine();

            }
        }

        static public void Main(string[] args)
        {
            if (args.Length == 0)
            {
                var sw = new StreamWriter(Console.OpenStandardOutput()) { AutoFlush = false };
                Console.SetOut(sw);
            }
            Solve();
            Console.Out.Flush();
        }
        static Random rand = new Random();
        static class Console_
        {
            private static Queue<string> param = new Queue<string>();
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            public static string NextString()
            {
                if (param.Count == 0) foreach (var item in Console.ReadLine().Split(' ')) param.Enqueue(item);
                return param.Dequeue();
            }
        }
        static int NextInt => int.Parse(Console_.NextString());
        static long NextLong => long.Parse(Console_.NextString());
        static double NextDouble => double.Parse(Console_.NextString());
        static string NextString => Console_.NextString();
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        static List<int> NextIntList(int N) => Enumerable.Repeat(0, N).Select(_ => NextInt).ToList();
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        static List<long> NextLongList(int N) => Enumerable.Repeat(0, N).Select(_ => NextLong).ToList();
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        static List<double> NextDoubleList(int N) => Enumerable.Repeat(0, N).Select(_ => NextDouble).ToList();
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        static List<string> NextStringList(int N) => Enumerable.Repeat(0, N).Select(_ => NextString).ToList();
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        static IEnumerable<T> OrderByRand<T>(this IEnumerable<T> x) => x.OrderBy(_ => rand.Next());
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        static IEnumerable<T> Repeat<T>(T v, long n) => Enumerable.Repeat<T>(v, (int)n);
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        static List<T> Sort<T>(List<T> l) where T : IComparable
        {
            var tmp = l.ToArray(); Array.Sort(tmp); return tmp.ToList();
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        static List<T> Sort<T>(List<T> l, Comparison<T> comp) where T : IComparable
        {
            var tmp = l.ToArray(); Array.Sort(tmp, comp); return tmp.ToList();
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        static List<T> RevSort<T>(List<T> l) where T : IComparable
        {
            var tmp = l.ToArray(); Array.Sort(tmp, (x, y) => y.CompareTo(x)); return tmp.ToList();
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        static List<T> RevSort<T>(List<T> l, Comparison<T> comp) where T : IComparable
        {
            var tmp = l.ToArray(); Array.Sort(tmp, (x, y) => comp(y, x)); return tmp.ToList();
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        static IEnumerable<long> Prime(long x)
        {
            if (x < 2) yield break;
            yield return 2;
            var halfx = x / 2;
            var table = new bool[halfx + 1];
            var max = (long)(Math.Sqrt(x) / 2);
            for (long i = 1; i <= max; ++i)
            {
                if (table[i]) continue;
                var add = 2 * i + 1;
                yield return add;
                for (long j = 2 * i * (i + 1); j <= halfx; j += add) table[j] = true;
            }
            for (long i = max + 1; i <= halfx; ++i) if (!table[i] && 2 * i + 1 <= x) yield return 2 * i + 1;
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        static IEnumerable<long> Divisor(long x)
        {
            if (x < 1) yield break;
            var max = (long)Math.Sqrt(x);
            for (long i = 1; i <= max; ++i)
            {
                if (x % i != 0) continue;
                yield return i;
                if (i != x / i) yield return x / i;
            }
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        static long GCD(long a, long b)
        {
            while (b > 0) { var tmp = b; b = a % b; a = tmp; }
            return a;
        }
        class PQ<T> where T : IComparable
        {
            private List<T> h;
            private Comparison<T> c;
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            public PQ(int cap, Comparison<T> c, bool asc = true) { h = new List<T>(cap); this.c = asc ? c : (x, y) => c(y, x); }
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            public PQ(Comparison<T> c, bool asc = true) { h = new List<T>(); this.c = asc ? c : (x, y) => c(y, x); }
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            public PQ(int cap, bool asc = true) : this(cap, (x, y) => x.CompareTo(y), asc) { }
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            public PQ(bool asc = true) : this((x, y) => x.CompareTo(y), asc) { }
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            public void Push(T v)
            {
                var i = h.Count;
                h.Add(v);
                while (i > 0)
                {
                    var ni = (i - 1) / 2;
                    if (c(v, h[ni]) >= 0) break;
                    h[i] = h[ni]; i = ni;
                }
                h[i] = v;
            }
            public T Peek => h[0];
            public int Count => h.Count;
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            public T Pop()
            {
                var r = h[0];
                var v = h[h.Count - 1];
                h.RemoveAt(h.Count - 1);
                if (h.Count == 0) return r;
                var i = 0;
                while (i * 2 + 1 < h.Count)
                {
                    var i1 = i * 2 + 1;
                    var i2 = i * 2 + 2;
                    if (i2 < h.Count && c(h[i1], h[i2]) > 0) i1 = i2;
                    if (c(v, h[i1]) <= 0) break;
                    h[i] = h[i1]; i = i1;
                }
                h[i] = v;
                return r;
            }
        }
        class PQ<TKey, TValue> where TKey : IComparable
        {
            private PQ<Tuple<TKey, TValue>> q;
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            public PQ(int cap, Comparison<TKey> c, bool asc = true) { q = new PQ<Tuple<TKey, TValue>>(cap, (x, y) => c(x.Item1, y.Item1), asc); }
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            public PQ(Comparison<TKey> c, bool asc = true) { q = new PQ<Tuple<TKey, TValue>>((x, y) => c(x.Item1, y.Item1), asc); }
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            public PQ(int cap, bool asc = true) : this(cap, (x, y) => x.CompareTo(y), asc) { }
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            public PQ(bool asc = true) : this((x, y) => x.CompareTo(y), asc) { }
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            public void Push(TKey k, TValue v) => q.Push(Tuple.Create(k, v));
            public Tuple<TKey, TValue> Peek => q.Peek;
            public int Count => q.Count;
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            public Tuple<TKey, TValue> Pop() => q.Pop();
        }
        class Mod
        {
            static public long _mod = 1000000007;
            private long _val = 0;
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            public Mod() { }
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            public Mod(long x) { if (x < _mod && x >= 0) _val = x; else if ((_val = x % _mod) < 0) _val += _mod; }
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            static public implicit operator Mod(long x) => new Mod(x);
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            static public implicit operator long(Mod x) => x._val;
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            static public Mod operator +(Mod x, Mod y) { var t = x._val + y._val; return new Mod() { _val = t < _mod ? t : t - _mod }; }
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            static public Mod operator +(Mod x, long y) => x._val + y;
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            static public Mod operator +(long x, Mod y) => x + y._val;
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            static public Mod operator -(Mod x, Mod y) { var t = x._val - y._val; return new Mod() { _val = t < _mod ? t : t - _mod }; }
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            static public Mod operator -(Mod x, long y) => x._val - y;
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            static public Mod operator -(long x, Mod y) => x - y._val;
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            static public Mod operator /(Mod x, Mod y) => x._val * Inverse(y._val);
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            static public bool operator ==(Mod x, Mod y) => x._val == y._val;
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            static public bool operator ==(Mod x, long y) => x._val == y;
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            static public bool operator ==(long x, Mod y) => x == y._val;
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            static public bool operator !=(Mod x, Mod y) => x._val != y._val;
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            static public bool operator !=(Mod x, long y) => x._val != y;
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            static public bool operator !=(long x, Mod y) => x != y._val;
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            static public Mod Pow(Mod x, long y)
            {
                Mod a = 1;
                while (y != 0)
                {
                    if ((y & 1) == 1) a *= x;
                    x *= x; y >>= 1;
                }
                return a;
            }
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            static public long Inverse(long x)
            {
                long b = _mod, r = 1, u = 0, t = 0;
                while (b > 0)
                {
                    var q = x / b;
                    t = u; u = r - q * u; r = t;
                    t = b; b = x - q * b; x = t;
                }
                return r < 0 ? r + _mod : r;
            }
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            override public bool Equals(object obj) => obj == null ? false : _val == ((Mod)obj)._val;
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            public bool Equals(Mod obj) => obj == null ? false : _val == obj._val;
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            public override int GetHashCode() => _val.GetHashCode();
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            public override string ToString() => _val.ToString();
            static private List<Mod> _fact = new List<Mod>();
            static private List<Mod> _ifact = new List<Mod>();
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            static private void Build(int n)
            {
                if (n >= _fact.Count) for (int i = _fact.Count; i <= n; ++i)
                        if (i == 0L) { _fact.Add(1); _ifact.Add(1); }
                        else { _fact.Add(_fact[i - 1] * i); _ifact.Add(_ifact[i - 1] * Mod.Pow(i, _mod - 2)); }
            }
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            static public Mod Comb(int n, int k)
            {
                Build(n);
                if (n == 0 && k == 0) return 1;
                if (n < k || n < 0) return 0;
                return _ifact[n - k] * _ifact[k] * _fact[n];
            }
        }
        class Mat<T>
        {
            private T[,] m;
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            public Mat(T[,] v) { m = (T[,])v.Clone(); }
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            static public implicit operator Mat<T>(T[,] v) => new Mat<T>(v);
            public T this[int r, int c] { get { return m[r, c]; } set { m[r, c] = value; } }
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            static public Mat<T> operator +(Mat<T> a, T x)
            {
                var tm = (T[,])a.m.Clone();
                for (int r = 0; r < tm.GetLength(0); ++r) for (int c = 0; c < tm.GetLength(1); ++c) tm[r, c] += (dynamic)x;
                return new Mat<T>(tm);
            }
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            static public Mat<T> operator +(Mat<T> a, Mat<T> b)
            {
                var tm = (T[,])a.m.Clone();
                for (int r = 0; r < tm.GetLength(0); ++r) for (int c = 0; c < tm.GetLength(1); ++c) tm[r, c] += (dynamic)b[r, c];
                return new Mat<T>(tm);
            }
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            static public Mat<T> operator -(Mat<T> a, T x)
            {
                var tm = (T[,])a.m.Clone();
                for (int r = 0; r < tm.GetLength(0); ++r) for (int c = 0; c < tm.GetLength(1); ++c) tm[r, c] -= (dynamic)x;
                return new Mat<T>(tm);
            }
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            static public Mat<T> operator -(Mat<T> a, Mat<T> b)
            {
                var tm = (T[,])a.m.Clone();
                for (int r = 0; r < tm.GetLength(0); ++r) for (int c = 0; c < tm.GetLength(1); ++c) tm[r, c] -= (dynamic)b[r, c];
                return new Mat<T>(tm);
            }
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            static public Mat<T> operator *(Mat<T> a, T x)
            {
                var tm = (T[,])a.m.Clone();
                for (int r = 0; r < tm.GetLength(0); ++r) for (int c = 0; c < tm.GetLength(1); ++c) tm[r, c] *= (dynamic)x;
                return new Mat<T>(tm);
            }
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            static public Mat<T> operator *(Mat<T> a, Mat<T> b)
            {
                var nr = a.m.GetLength(0); var nc = b.m.GetLength(1);
                var tm = new T[nr, nc];
                for (int i = 0; i < nr; ++i) for (int j = 0; j < nc; ++j) tm[i, j] = (dynamic)0;
                for (int r = 0; r < nr; ++r) for (int c = 0; c < nc; ++c) for (int i = 0; i < a.m.GetLength(1); ++i) tm[r, c] += a[r, i] * (dynamic)b[i, c];
                return new Mat<T>(tm);
            }
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            static public Mat<T> Pow(Mat<T> x, long y)
            {
                var n = x.m.GetLength(0);
                var t = (Mat<T>)new T[n, n];
                for (int i = 0; i < n; ++i) for (int j = 0; j < n; ++j) t[i, j] = (dynamic)(i == j ? 1 : 0);
                while (y != 0)
                {
                    if ((y & 1) == 1) t *= x;
                    x *= x; y >>= 1;
                }
                return t;
            }
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        static Mat<T> Pow<T>(Mat<T> x, long y) => Mat<T>.Pow(x, y);
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        static Mod Pow(Mod x, long y) => Mod.Pow(x, y);
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        static long Pow(long x, long y)
        {
            long a = 1;
            while (y != 0)
            {
                if ((y & 1) == 1) a *= x;
                x *= x; y >>= 1;
            }
            return a;
        }
        static List<long> _fact = new List<long>();
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        static void _Build(int n)
        {
            if (n >= _fact.Count) for (int i = _fact.Count; i <= n; ++i)
                    if (i == 0L) _fact.Add(1);
                    else _fact.Add(_fact[i - 1] * i);
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        static long Comb(int n, int k)
        {
            _Build(n);
            if (n == 0 && k == 0) return 1;
            if (n < k || n < 0) return 0;
            return _fact[n] / _fact[k] / _fact[n - k];
        }
    }
}