Problem:
Score : 600 points

Problem Statement

We have N strings of lowercase English letters: S_1, S_2, \cdots, S_N.

Takahashi wants to make a string that is a palindrome by choosing one or more of these strings - the same string can be chosen more than once - and concatenating them in some order of his choice.

The cost of using the string S_i once is C_i, and the cost of using it multiple times is C_i multiplied by that number of times.

Find the minimum total cost needed to choose strings so that Takahashi can make a palindrome.

If there is no choice of strings in which he can make a palindrome, print -1.

Constraints

1 \leq N \leq 50

1 \leq |S_i| \leq 20

S_i consists of lowercase English letters.

1 \leq C_i \leq 10^9

Input

Input is given from Standard Input in the following format:

N
S_1 C_1
S_2 C_2
:
S_N C_N

Output

Print the minimum total cost needed to choose strings so that Takahashi can make a palindrome, or -1 if there is no such choice.

Sample Input 1

3
ba 3
abc 4
cbaa 5

Sample Output 1

7

We have ba, abc, and cbaa.

For example, we can use ba once and abc once for a cost of 7, then concatenate them in the order abc, ba to make a palindrome.
Also, we can use abc once and cbaa once for a cost of 9, then concatenate them in the order cbaa, abc to make a palindrome.

We cannot make a palindrome for a cost less than 7, so we should print 7.

Sample Input 2

2
abcab 5
cba 3

Sample Output 2

11

We can choose abcab once and cba twice, then concatenate them in the order abcab, cba, cba to make a palindrome for a cost of 11.

Sample Input 3

4
ab 5
cba 3
a 12
ab 10

Sample Output 3

8

We can choose a once, which is already a palindrome, but it is cheaper to concatenate ab and cba.

Sample Input 4

2
abc 1
ab 2

Sample Output 4

-1

We cannot make a palindrome, so we should print -1.

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

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

namespace Program
{
    public static class ProblemF
    {
        static bool SAIKI = false;
        static public int numberOfRandomCases = 0;
        static public void MakeTestCase(List<string> _input, List<string> _output, ref Func<string[], bool> _outputChecker)
        {
        }
        static public void Solve()
        {
            var N = NN;
            var SList = Repeat(0, N).Select(_ => new {S = NS, C = NN}).ToArray();
            var q = new LIB_PriorityQueue<long, (long, bool)>();
            var prefixNokoriKaibun = new Dictionary<long, long>();
            var sufixNokoriKaibun = new Dictionary<long, long>();
            var prefix = new LIB_Dict<long, List<(long, long)>>(_ => new List<(long, long)>());
            var sufix = new LIB_Dict<long, List<(long, long)>>(_ => new List<(long, long)>());
            foreach(var item in SList) {
                var hash2 = new LIB_RollingHash(item.S);
                q.Push(item.C, (hash2, true));
                for(int i = 0;i < item.S.Length;++i) {
                    var hash = hash2.GetHash(0, i + 1);
                    prefix[hash].Add((hash2.GetHash(i + 1, item.S.Length), item.C));
                    var ok = true;
                    for(int j = i + 1;j < item.S.Length;++j) {
                        var dif = j - i;
                        if(item.S[j] != item.S[item.S.Length - dif]) {
                            ok = false;
                            break;
                        }
                    }
                    if(ok) {
                        if(prefixNokoriKaibun.ContainsKey(hash)) {
                            prefixNokoriKaibun[hash] = Min(prefixNokoriKaibun[hash], item.C);
                        }else {
                            prefixNokoriKaibun[hash] = item.C;
                        }
                    }
                }
                hash2 = new LIB_RollingHash(new string(item.S.Reverse().ToArray()));
                q.Push(item.C, (hash2, false));
                for(int i = item.S.Length-1;i >=0;--i) {
                    var hash = hash2.GetHash(0, item.S.Length - i);
                    sufix[hash].Add((hash2.GetHash(item.S.Length - i, item.S.Length), item.C));
                    var ok = true;
                    for(int j = i - 1;j >=0;--j) {
                        var dif = i - 1 - j;
                        if(item.S[j] != item.S[dif]) {
                            ok = false;
                            break;
                        }
                    }
                    if(ok) {
                        if(sufixNokoriKaibun.ContainsKey(hash)) {
                            sufixNokoriKaibun[hash] = Min(sufixNokoriKaibun[hash], item.C);
                        }else {
                            sufixNokoriKaibun[hash] = item.C;
                        }
                    }
                }
            }
            var ans = long.MaxValue;
            while(q.Count > 0) {
                var str = q.Pop();
                if(str.Value.Item2) {
                    if(sufixNokoriKaibun.ContainsKey(str.Value.Item1)) {
                        ans = Min(ans, str.Key + sufixNokoriKaibun[str.Value.Item1]);
                        continue;
                    }
                    foreach(var item in sufix[str.Value.Item1]) {
                        q.Push(str.Key + item.Item2, (item.Item1, false));
                    }
                }else {
                    if(prefixNokoriKaibun.ContainsKey(str.Value.Item1)) {
                        ans = Min(ans, str.Key + prefixNokoriKaibun[str.Value.Item1]);
                        continue;
                    }
                    foreach(var item in prefix[str.Value.Item1]) {
                        q.Push(str.Key + item.Item2, (item.Item1, true));
                    }
                }
            }
            if(ans == long.MaxValue)System.Console.WriteLine(-1);
            else System.Console.WriteLine(ans);
        }
        class Printer : StreamWriter
        {
            public override IFormatProvider FormatProvider { get { return CultureInfo.InvariantCulture; } }
            public Printer(Stream stream) : base(stream, new UTF8Encoding(false, true)) { base.AutoFlush = false; }
            public Printer(Stream stream, Encoding encoding) : base(stream, encoding) { base.AutoFlush = false; }
        }
        static LIB_FastIO fastio = new LIB_FastIODebug();
        static public void Main(string[] args) { if (args.Length == 0) { fastio = new LIB_FastIO(); Console.SetOut(new Printer(Console.OpenStandardOutput())); } if (SAIKI) { var t = new Thread(Solve, 134217728); t.Start(); t.Join(); } else Solve(); Console.Out.Flush(); }
        static long NN => fastio.Long();
        static double ND => fastio.Double();
        static string NS => fastio.Scan();
        static long[] NNList(long N) => Repeat(0, N).Select(_ => NN).ToArray();
        static double[] NDList(long N) => Repeat(0, N).Select(_ => ND).ToArray();
        static string[] NSList(long N) => Repeat(0, N).Select(_ => NS).ToArray();
        static long Count<T>(this IEnumerable<T> x, Func<T, bool> pred) => Enumerable.Count(x, pred);
        static IEnumerable<T> Repeat<T>(T v, long n) => Enumerable.Repeat<T>(v, (int)n);
        static IEnumerable<int> Range(long s, long c) => Enumerable.Range((int)s, (int)c);
        static IOrderedEnumerable<T> OrderByRand<T>(this IEnumerable<T> x) => Enumerable.OrderBy(x, _ => xorshift);
        static IOrderedEnumerable<T> OrderBy<T>(this IEnumerable<T> x) => Enumerable.OrderBy(x.OrderByRand(), e => e);
        static IOrderedEnumerable<T1> OrderBy<T1, T2>(this IEnumerable<T1> x, Func<T1, T2> selector) => Enumerable.OrderBy(x.OrderByRand(), selector);
        static IOrderedEnumerable<T> OrderByDescending<T>(this IEnumerable<T> x) => Enumerable.OrderByDescending(x.OrderByRand(), e => e);
        static IOrderedEnumerable<T1> OrderByDescending<T1, T2>(this IEnumerable<T1> x, Func<T1, T2> selector) => Enumerable.OrderByDescending(x.OrderByRand(), selector);
        static IOrderedEnumerable<string> OrderBy(this IEnumerable<string> x) => x.OrderByRand().OrderBy(e => e, StringComparer.OrdinalIgnoreCase);
        static IOrderedEnumerable<T> OrderBy<T>(this IEnumerable<T> x, Func<T, string> selector) => x.OrderByRand().OrderBy(selector, StringComparer.OrdinalIgnoreCase);
        static IOrderedEnumerable<string> OrderByDescending(this IEnumerable<string> x) => x.OrderByRand().OrderByDescending(e => e, StringComparer.OrdinalIgnoreCase);
        static IOrderedEnumerable<T> OrderByDescending<T>(this IEnumerable<T> x, Func<T, string> selector) => x.OrderByRand().OrderByDescending(selector, StringComparer.OrdinalIgnoreCase);
        static string Join<T>(this IEnumerable<T> x, string separator = "") => string.Join(separator, x);
        static uint xorshift { get { _xsi.MoveNext(); return _xsi.Current; } }
        static IEnumerator<uint> _xsi = _xsc();
        static IEnumerator<uint> _xsc() { uint x = 123456789, y = 362436069, z = 521288629, w = (uint)(DateTime.Now.Ticks & 0xffffffff); while (true) { var t = x ^ (x << 11); x = y; y = z; z = w; w = (w ^ (w >> 19)) ^ (t ^ (t >> 8)); yield return w; } }
    }
}
namespace Library {
    class LIB_FastIO
    {
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public LIB_FastIO() { str = Console.OpenStandardInput(); }
        readonly Stream str;
        readonly byte[] buf = new byte[1024];
        int len, ptr;
        public bool isEof = false;
        public bool IsEndOfStream
        {
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            get { return isEof; }
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        byte read()
        {
            if (isEof) throw new EndOfStreamException();
            if (ptr >= len)
            {
                ptr = 0;
                if ((len = str.Read(buf, 0, 1024)) <= 0)
                {
                    isEof = true;
                    return 0;
                }
            }
            return buf[ptr++];
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        char Char()
        {
            byte b = 0;
            do b = read();
            while (b < 33 || 126 < b);
            return (char)b;
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        virtual public string Scan()
        {
            var sb = new StringBuilder();
            for (var b = Char(); b >= 33 && b <= 126; b = (char)read())
                sb.Append(b);
            return sb.ToString();
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        virtual public long Long()
        {
            long ret = 0; byte b = 0; var ng = false;
            do b = read();
            while (b != '-' && (b < '0' || '9' < b));
            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';
            }
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        virtual public double Double() { return double.Parse(Scan(), CultureInfo.InvariantCulture); }
    }
    class LIB_FastIODebug : LIB_FastIO
    {
        Queue<string> param = new Queue<string>();
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        string NextString() { if (param.Count == 0) foreach (var item in Console.ReadLine().Split(' ')) param.Enqueue(item); return param.Dequeue(); }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public LIB_FastIODebug() { }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public override string Scan() => NextString();
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public override long Long() => long.Parse(NextString());
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public override double Double() => double.Parse(NextString());
    }
    class LIB_RollingHash
    {
        const ulong MASK30 = (1UL << 30) - 1;
        const ulong MASK31 = (1UL << 31) - 1;
        const ulong MOD = (1UL << 61) - 1;
        const ulong POSITIVIZER = MOD * ((1UL << 3) - 1);
        static readonly uint baseNum;
        static readonly List<ulong> basePow;
        List<ulong> hash;
        static LIB_RollingHash()
        {
            baseNum = (uint)new Random().Next(10000000, int.MaxValue);
            basePow = new ulong[] { 1 }.ToList();
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        static public implicit operator long(LIB_RollingHash x) => (long)x.hash.Last();
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public LIB_RollingHash(string s = "") : this(s.Select(e => (long)e)) { }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public LIB_RollingHash(IEnumerable<long> ary) { hash = new ulong[1].ToList(); foreach (var item in ary) Add(item); }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public void Add(long val)
        {
            hash.Add(Mod(Mul(hash.Last(), baseNum) + (ulong)val));
            basePow.Add(Mod(Mul(basePow.Last(), baseNum)));
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public long GetHash(int l, int r) => (long)Mod(hash[r] + POSITIVIZER - Mul(hash[l], basePow[r - l]));
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        static ulong Mul(ulong l, ulong r)
        {
            var lu = l >> 31;
            var ld = l & MASK31;
            var ru = r >> 31;
            var rd = r & MASK31;
            var middleBit = ld * ru + lu * rd;
            return ((lu * ru) << 1) + ld * rd + ((middleBit & MASK30) << 31) + (middleBit >> 30);
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        static ulong Mul(ulong l, uint r)
        {
            var rd = r & MASK31;
            var middleBit = (l >> 31) * rd;
            return (l & MASK31) * rd + ((middleBit & MASK30) << 31) + (middleBit >> 30);
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        static ulong Mod(ulong val)
        {
            val = (val & MOD) + (val >> 61);
            return val < MOD ? val : val - MOD;
        }
    }
    class LIB_Dict<K, V> : Dictionary<K, V>
    {
        Func<K, V> d;
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public LIB_Dict(Func<K, V> _d) { d = _d; }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public LIB_Dict() : this(_ => default(V)) { }
        new public V this[K i]
        {
            get
            {
                return TryGetValue(i, out var v) ? v : base[i] = d(i);
            }
            set
            {
                base[i] = value;
            }
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public override int GetHashCode()
        {
            unchecked
            {
                var ret = 23;
                for (var e = GetEnumerator(); e.MoveNext();) ret = (ret * 37) ^ (e.Current.Key.GetHashCode() << 5 + e.Current.Key.GetHashCode()) ^ e.Current.Value.GetHashCode();
                return ret;
            }
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public override bool Equals(object obj)
        {
            var o = (LIB_Dict<K, V>)obj;
            foreach (var kv in o)
            {
                if (!ContainsKey(kv.Key) || !this[kv.Key].Equals(kv.Value))
                    return false;
            }
            return o.Count == Count;
        }
    }
    class LIB_PriorityQueue<T>
    {
        List<T> heap;
        Comparison<T> comp;
        public T Peek => heap[0];
        public long Count => heap.Count;
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public LIB_PriorityQueue(long cap, Comparison<T> cmp, bool asc = true)
        {
            heap = new List<T>((int)cap);
            comp = asc ? cmp : (x, y) => cmp(y, x);
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public LIB_PriorityQueue(Comparison<T> cmp, bool asc = true)
        {
            heap = new List<T>();
            comp = asc ? cmp : (x, y) => cmp(y, x);
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public LIB_PriorityQueue(long cap, bool asc = true) : this(cap, Comparer<T>.Default.Compare, asc) { }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public LIB_PriorityQueue(bool asc = true) : this(Comparer<T>.Default.Compare, asc) { }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public void Push(T val)
        {
            var i = heap.Count;
            heap.Add(val);
            while (i > 0)
            {
                var ni = (i - 1) / 2;
                if (comp(val, heap[ni]) >= 0) break;
                heap[i] = heap[ni];
                i = ni;
            }
            heap[i] = val;
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public T Pop()
        {
            var ret = heap[0];
            var val = heap[heap.Count - 1];
            heap.RemoveAt(heap.Count - 1);
            if (heap.Count == 0) return ret;
            var i = 0; while ((i << 1) + 1 < heap.Count)
            {
                var i1 = (i << 1) + 1;
                var i2 = (i << 1) + 2;
                if (i2 < heap.Count && comp(heap[i1], heap[i2]) > 0) i1 = i2;
                if (comp(val, heap[i1]) <= 0) break;
                heap[i] = heap[i1]; i = i1;
            }
            heap[i] = val;
            return ret;
        }
    }
    class LIB_PriorityQueue<TK, TV>
    {
        LIB_PriorityQueue<KeyValuePair<TK, TV>> q;
        public KeyValuePair<TK, TV> Peek => q.Peek;
        public long Count => q.Count;
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public LIB_PriorityQueue(long cap, Comparison<TK> cmp, bool asc = true)
        {
            q = new LIB_PriorityQueue<KeyValuePair<TK, TV>>(cap, (x, y) => cmp(x.Key, y.Key), asc);
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public LIB_PriorityQueue(Comparison<TK> cmp, bool asc = true)
        {
            q = new LIB_PriorityQueue<KeyValuePair<TK, TV>>((x, y) => cmp(x.Key, y.Key), asc);
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public LIB_PriorityQueue(long cap, bool asc = true) : this(cap, Comparer<TK>.Default.Compare, asc) { }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public LIB_PriorityQueue(bool asc = true) : this(Comparer<TK>.Default.Compare, asc) { }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public void Push(TK k, TV v) => q.Push(new KeyValuePair<TK, TV>(k, v));
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public KeyValuePair<TK, TV> Pop() => q.Pop();
    }
}