Problem:
Score : 300 points

Problem Statement

How many integer sequences A_1,A_2,\ldots,A_N of length N satisfy all of the following conditions?

0 \leq A_i \leq 9

There exists some i such that A_i=0 holds.

There exists some i such that A_i=9 holds.

The answer can be very large, so output it modulo 10^9 + 7.

Constraints

1 \leq N \leq 10^6

N is an integer.

Input

Input is given from Standard Input in the following format:

N

Output

Print the answer modulo 10^9 + 7.

Sample Input 1

2

Sample Output 1

2

Two sequences \{0,9\} and \{9,0\} satisfy all conditions.

Sample Input 2

1

Sample Output 2

0

Sample Input 3

869121

Sample Output 3

2511445

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

using System;
using System.Collections.Generic;
using System.Linq;
using System.IO;
using System.Text;
using System.Numerics;
using System.Threading;
using System.Runtime.CompilerServices;
using System.Diagnostics;
using static System.Math;
using static System.Array;
using static AtCoder.Cout;
using static AtCoder.Tool;
using static AtCoder.Graph;
using static AtCoder.ModInt;
namespace AtCoder
{
    class AC
    {
        const int MOD = 1000000007;
        //const int MOD = 998244353;
        const int INF = int.MaxValue / 2;
        const long SINF = long.MaxValue / 3;
        static readonly int[] dI = { 0, 1, 0, -1, 1, -1, -1, 1 };
        static readonly int[] dJ = { 1, 0, -1, 0, 1, 1, -1, -1 };
        static void Main(string[] args)
        {
            //var sw = new StreamWriter(Console.OpenStandardOutput()) { AutoFlush = false }; Console.SetOut(sw);

            /*var th = new Thread(Run, 1 << 26);
            th.Start();
            th.Join();*/

            Run();
            Console.Out.Flush();
        }
        static void Run()
        {
            int Testcase = 1;
            //Testcase = Cin.Int;
            for (var _ = 0; _ < Testcase; _++) Solve();
        }
        static void Solve()
        {
            var N = Cin.Int;
            ModInt ans = 0;
            ans = Modpow(10, N);
            ans -= 2 * Modpow(9, N);
            ans += Modpow(8, N);
            OutL(ans.value);
        }
    }
    public struct Edge
    {
        public int from, to;
        public long w;
        public Edge(int to, long weight) { this.to = to; w = weight; from = -1; }
        public Edge(int from, int to, long weight) { this.from = from; this.to = to; w = weight; }
    }
    struct ModInt
    {
        public long value;
        const int MOD = 1000000007;
        //const int MOD = 998244353;
        public ModInt(long value) { this.value = value; }
        public static implicit operator ModInt(long a)
        {
            var ret = a % MOD;
            return new ModInt(ret < 0 ? (ret + MOD) : ret);
        }
        public static ModInt operator +(ModInt a, ModInt b) => (a.value + b.value);
        public static ModInt operator -(ModInt a, ModInt b) => (a.value - b.value);
        public static ModInt operator *(ModInt a, ModInt b) => (a.value * b.value);
        public static ModInt operator /(ModInt a, ModInt b) => a * Modpow(b, MOD - 2);

        public static ModInt operator <<(ModInt a, int n) => (a.value << n);
        public static ModInt operator >>(ModInt a, int n) => (a.value >> n);
        public static ModInt operator ++(ModInt a) => a.value + 1;
        public static ModInt operator --(ModInt a) => a.value - 1;
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public static ModInt Modpow(ModInt a, long n)
        {
            if (n == 0) return 1;
            if (n < 0) return Modpow(Modpow(a, -n), MOD - 2);
            var k = a;
            ModInt ret = 1;
            while (n > 0)
            {
                if ((n & 1) != 0) ret *= k;
                k *= k;
                n >>= 1;
            }
            return ret;
        }
        private static readonly List<long> Factorials = new List<long>() { 1 };
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public static ModInt Fac(int n)
        {
            for (var i = Factorials.Count(); i <= n; i++)
            {
                Factorials.Add((Factorials[i - 1] * i) % MOD);
            }
            return Factorials[n];
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public static ModInt nCr(int n, int r)
        {
            if (n < 0 || r < 0) return 0;
            return n < r ? 0 : Fac(n) / (Fac(r) * Fac(n - r));
        }
        public static explicit operator int(ModInt a) => (int)a.value;
    }
    public class SCC
    {
        int n;
        struct Edge_S { public int from, to; public Edge_S(int f, int t) { from = f; to = t; } }
        List<Edge_S> E;
        int[] id;
        public SCC(int size) { n = size; E = new List<Edge_S>(); }
        public void Add_Edge(int from, int to)
        {
            E.Add(new Edge_S(from, to));
        }
        public int[][] Scc_Result()
        {
            var start = new int[n + 1];
            var nxt = new int[E.Count];
            foreach (var ed in E) start[ed.from + 1]++;
            for (var i = 0; i < n; i++) start[i + 1] += start[i];
            var itr = new int[n + 1]; for (var i = 0; i <= n; i++) itr[i] = start[i];
            foreach (var ed in E) nxt[itr[ed.from]++] = ed.to;

            int now = 0;
            int[] ord = new int[n];
            int[] low = new int[n];
            id = new int[n];
            var V = new Stack<int>();
            for (var i = 0; i < n; i++) ord[i] = -1;
            int nowid = 0;

            Action<int> DFS = null;
            DFS = (v) =>
            {
                low[v] = ord[v] = now++;
                V.Push(v);
                for (var i = start[v]; i < start[v + 1]; i++)
                {
                    var nx = nxt[i];
                    if (ord[nx] == -1)
                    {
                        DFS(nx);
                        low[v] = Min(low[v], low[nx]);
                    }
                    else
                    {
                        low[v] = Min(low[v], ord[nx]);
                    }
                }
                if (low[v] == ord[v])
                {
                    while (true)
                    {
                        var u = V.Pop();
                        id[u] = nowid;
                        ord[u] = n + 1;
                        if (u == v) break;
                    }
                    nowid++;
                }
            };
            for (var i = 0; i < n; i++) if (ord[i] == -1) DFS(i);
            for (var i = 0; i < n; i++)
            {
                id[i] = nowid - 1 - id[i];
                itr[i] = 0;
            }
            var scc = new int[nowid][];
            for (var i = 0; i < n; i++) itr[id[i]]++;
            for (var i = 0; i < nowid; i++) scc[i] = new int[itr[i]];
            for (var i = 0; i < n; i++) scc[id[i]][--itr[id[i]]] = i;
            return scc;
        }
        public int v_id(int v) => id[v];
    }
    public class Two_SAT
    {
        // use with SCC Library
        int n;
        bool[] result;
        SCC scc;
        readonly int md;
        public Two_SAT(int size)
        {
            n = size;
            result = new bool[n];
            scc = new SCC(n << 1);
            md = n << 1;
        }
        public void Add_Closure(int i, int j)
        {
            scc.Add_Edge((i + n) % md, j);
            scc.Add_Edge((j + n) % md, i);
        }
        public bool Satisfy()
        {
            scc.Scc_Result();
            for (var i = 0; i < n; i++)
            {
                int j = scc.v_id(i), k = scc.v_id(i + n);
                if (j == k) return false;
                result[i] = j > k;
            }
            return true;
        }
        public bool[] ans() => result;
    }
    public class Dinic
    {
        readonly int n;
        const int inf = int.MaxValue / 2;
        public class Edge_F
        {
            public int _to { get; set; }
            public long _cap { get; set; }
            public int _rev { get; set; }
            public Edge_F(int to, long cap, int rev) { _to = to; _cap = cap; _rev = rev; }
        }
        List<List<Edge_F>> G;
        int[] level, itr;
        public Dinic(int vertice)
        {
            n = vertice;
            level = new int[n];
            itr = new int[n];
            G = new List<List<Edge_F>>();
            for (var _ = 0; _ < n; _++) G.Add(new List<Edge_F>());
        }
        /*================ ^ _ ^ ==================*/
        //辺の追加（from->to,容量cap）
        public void Add_Edge(int from, int to, long cap)
        {
            G[from].Add(new Edge_F(to, cap, G[to].Count()));
            G[to].Add(new Edge_F(from, 0, G[from].Count() - 1));
        }
        //bfsパート（levelの設定）
        void Bfs(int s)
        {
            //Fillはバージョン古いと使えないため...
            for (var i = 0; i < n; i++) level[i] = -1;
            level[s] = 0;
            var Q = new Queue<int>();
            Q.Enqueue(s);
            while (Q.Any())
            {
                int v = Q.Dequeue();
                foreach (var ed in G[v])
                {
                    if (ed._cap > 0 && level[ed._to] == -1)
                    {
                        level[ed._to] = level[v] + 1;
                        Q.Enqueue(ed._to);
                    }
                }
            }
        }
        //dfsパート（増加パスの探索）
        long Dfs(int v, int t, long f)
        {
            if (v == t) return f;
            for (var i = itr[v]; i < G[v].Count(); i++)
            {
                itr[v] = i;
                var ed = G[v][i];
                if (ed._cap > 0 && level[v] < level[ed._to])
                {
                    var d = Dfs(ed._to, t, Min(f, ed._cap));
                    if (d > 0)
                    {
                        ed._cap -= d;
                        G[ed._to][ed._rev]._cap += d;
                        return d;
                    }
                }
            }
            return 0;
        }
        //s->tの最大流を返す
        //一般:O(N^2M)
        //二部グラフマッチング:O(M*Sqrt(N))
        //辺の容量が全て同じ:O(min(n^{2/3},m^{1/2})*m)
        //になるらしい
        public long Max_Flow(int s, int t)
        {
            long ret = 0;
            for (; ; )
            {
                Bfs(s);
                if (level[t] == -1) return ret;
                for (var i = 0; i < n; i++) itr[i] = 0;
                var flow = 0L;
                do { ret += flow; flow = Dfs(s, t, inf); }
                while (flow > 0);
            }
        }
        //グラフの状況を返す
        public List<List<Edge_F>> GetGraph() => G;
    }
    public static class Graph
    {
        const long inf = long.MaxValue / 3;
        public static List<List<T>> Gen_Graph<T>(int size)
        {
            var ret = new List<List<T>>();
            for (var i = 0; i < size; i++) ret.Add(new List<T>());
            return ret;
        }
        public static long[] Dijkstra(List<List<Edge>> G, int st)
        {
            int N = G.Count();
            long[] ret = new long[N];
            var V = new Priority_Queue<Tuple<long, int>>((x, y) => Sig(x.Item1 - y.Item1));
            for (var i = 0; i < N; i++) ret[i] = inf;
            ret[st] = 0; V.Enqueue(new Tuple<long, int>(0, st));
            while (V.Any())
            {
                var cur = V.Dequeue();
                int v = cur.Item2;
                long cd = cur.Item1;
                if (ret[v] < cd) continue;
                foreach (var ed in G[v])
                {
                    if (ret[ed.to] > cd + ed.w)
                    {
                        ret[ed.to] = cd + ed.w;
                        V.Enqueue(new Tuple<long, int>(ret[ed.to], ed.to));
                    }
                }
            }
            return ret;
        }
        public static long[] Bellman_Frod(List<Edge> E, int st, int N, out bool neg_close)
        {
            var ret = new long[N];
            for (var i = 0; i < N; i++) ret[i] = inf;
            ret[st] = 0;
            for (var i = 0; i < N; i++)
            {
                foreach (var ed in E)
                {
                    if (ret[ed.from] != inf && ret[ed.to] > ret[ed.from] + ed.w)
                    {
                        if (i == N - 1) { neg_close = true; return ret; }
                        ret[ed.to] = ret[ed.from] + ed.w;
                    }
                }
            }
            neg_close = false;
            return ret;
        }
    }
    public class Priority_Queue<T>
    {
        private List<T> Q;
        private readonly Comparison<T> Func_Compare;
        public Priority_Queue(Comparison<T> comp)
        {
            Func_Compare = comp;
            Q = new List<T>();
        }
        private void PushHeap(T item)
        {
            int n = Q.Count();
            Q.Add(item);

            while (n != 0)
            {
                int pIndex = (n - 1) / 2;

                if (Func_Compare(Q[n], Q[pIndex]) < 0)
                {
                    Swap(n, pIndex);
                }
                else { break; }

                n = pIndex;
            }
        }
        private void PopHeap()
        {
            int n = Q.Count() - 1;
            Q[0] = Q[n];
            Q.RemoveAt(n);
            int cur = 0;
            int comp;
            while (2 * cur + 1 <= n - 1)
            {
                int c1 = 2 * cur + 1;
                int c2 = 2 * (cur + 1);
                if (c1 == n - 1)
                {
                    comp = c1;
                }
                else
                {

                    comp = Func_Compare(Q[c1], Q[c2]) < 0 ? c1 : c2;
                }

                if (Func_Compare(Q[cur], Q[comp]) > 0)
                {
                    Swap(cur, comp);
                }
                else { break; }

                cur = comp;
            }
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        private void Swap(int a, int b)
        {
            T keep = Q[a];
            Q[a] = Q[b];
            Q[b] = keep;
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public void Enqueue(T value) => PushHeap(value);
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public T Dequeue()
        {
            T ret = Q[0];
            PopHeap();
            return ret;
        }
        public T Peek() => Q[0];
        public int Count() => Q.Count();
        public bool Any() => Q.Any();
    }
    public class SegmentTree<T>
    {
        //1-indexed type
        int n;
        T[] Tree;
        Func<T, T, T> f;
        T ex;
        int L;
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public SegmentTree(int size, Func<T, T, T> fun, T exvalue)
        {
            ex = exvalue;
            f = fun;
            n = size;
            Tree = new T[n << 1];
            L = (n << 1) - 1;
            for (var i = 0; i <= L; i++) Tree[i] = ex;
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public SegmentTree(int size, Func<T, T, T> fun, T exvalue, T[] initial)
        {
            ex = exvalue;
            n = size;
            f = fun;
            Tree = new T[n << 1];
            L = (n << 1) - 1;
            for (var i = 0; i <= L; i++) Tree[i] = (n <= i && i <= n + initial.Length - 1) ? initial[i - n] : ex;
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public void Set_All()
        {
            for (var i = n - 1; i >= 1; i--) Tree[i] = f(Tree[i << 1], Tree[(i << 1) | 1]);
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public void Assign(int idx, T nxt) => Tree[idx + n] = nxt;
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public void Update(int idx)
        {
            int now = idx + n;
            while (now > 1)
            {
                now >>= 1;
                Tree[now] = f(Tree[now << 1], Tree[now << 1 | 1]);
            }
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public void Query_Update(int idx, T nxt)
        {
            Assign(idx, nxt); Update(idx);
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public void Query_Update_func(int idx, T y)
        {
            Assign(idx, f(Peek(idx), y)); Update(idx);
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public T Query_Fold(int l, int r)
        {
            int L = n + l;
            int R = n + r;
            T vL = ex, vR = ex;
            while (L < R)
            {
                if (L % 2 == 1)
                {
                    vL = f(vL, Tree[L]);
                    L++;
                }
                if (R % 2 == 1)
                {
                    vR = f(Tree[R - 1], vR);
                    R--;
                }
                L >>= 1;
                R >>= 1;
            }
            return f(vL, vR);
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public T Peek(int idx) => Tree[idx + n];
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public void Display(int len)
        {
            for (var i = 0; i < len; i++) Console.Write($"{Tree[i + n]} ");
            Console.WriteLine();
        }
    }
    public class UnionFind
    {
        private int[] parent;
        private int[] rank;
        private int[] size;
        public UnionFind(int n)
        {
            parent = new int[n];
            rank = new int[n];
            size = new int[n];
            for (var i = 0; i < n; i++)
            {
                parent[i] = i;
                rank[i] = 0;
                size[i] = 1;
            }
        }

        public int Root(int x)
        {
            return parent[x] == x ? x : parent[x] = Root(parent[x]);
        }

        public bool SameRoot(int x, int y)
        {
            return Root(x) == Root(y);
        }

        public void Unite(int x, int y)
        {
            x = Root(x);
            y = Root(y);
            if (x == y) { return; }

            if (rank[x] < rank[y])
            {
                parent[x] = y;
                size[y] += size[x];
                size[x] = 0;
            }
            else
            {
                parent[y] = x;
                if (rank[x] == rank[y]) { rank[x]++; }
                size[x] += size[y];
                size[y] = 0;
            }
        }

        public int SizeOf(int x)
        {
            return size[Root(x)];
        }

    }
    static class Cin
    {
        public static string[] ReadSplit => Console.ReadLine().Split();
        public static int[] ReadSplitInt => ConvertAll(ReadSplit, int.Parse);
        public static long[] ReadSplitLong => ConvertAll(ReadSplit, long.Parse);
        public static double[] ReadSplit_Double => ConvertAll(ReadSplit, double.Parse);
        public static string Str => Console.ReadLine();
        public static int Int => int.Parse(Console.ReadLine());
        public static long Long => long.Parse(Console.ReadLine());
        public static double Double => double.Parse(Console.ReadLine());
        public static T Conv<T>(string input)
        {
            if (typeof(T).Equals(typeof(ModInt)))
            {
                return (T)(dynamic)(long.Parse(input));
            }
            return (T)Convert.ChangeType(input, typeof(T));
        }
        public static void Scanf<T>(out T a) => a = Conv<T>(Console.ReadLine());
        public static void Scanf<T, U>(out T a, out U b)
        { var q = ReadSplit; a = Conv<T>(q[0]); b = Conv<U>(q[1]); }
        public static void Scanf<T, U, V>(out T a, out U b, out V c)
        { var q = ReadSplit; a = Conv<T>(q[0]); b = Conv<U>(q[1]); c = Conv<V>(q[2]); }
        public static void Scanf<T, U, V, W>(out T a, out U b, out V c, out W d)
        { var q = ReadSplit; a = Conv<T>(q[0]); b = Conv<U>(q[1]); c = Conv<V>(q[2]); d = Conv<W>(q[3]); }
        public static void Scanf<T, U, V, W, X>(out T a, out U b, out V c, out W d, out X e)
        { var q = ReadSplit; a = Conv<T>(q[0]); b = Conv<U>(q[1]); c = Conv<V>(q[2]); d = Conv<W>(q[3]); e = Conv<X>(q[4]); }
    }
    static class Cout
    {
        public static void OutL(object s) => Console.WriteLine(s);
        public static void Out_Sep<T>(IEnumerable<T> s) => Console.WriteLine(string.Join(" ", s));
        public static void Out_Sep<T>(IEnumerable<T> s, string sep) => Console.WriteLine(string.Join($"{sep}", s));
        public static void Out_Sep(params object[] s) => Console.WriteLine(string.Join(" ", s));
        public static void Out_One(object s) => Console.Write($"{s} ");
        public static void Out_One(object s, string sep) => Console.Write($"{s}{sep}");
        public static void Endl() => Console.WriteLine();
    }
    public static class Tool
    {
        static public void Initialize<T>(ref T[] array, T initialvalue)
        {
            array = ConvertAll(array, x => initialvalue);
        }
        static public void Swap<T>(ref T a, ref T b)
        {
            T keep = a;
            a = b;
            b = keep;
        }
        static public void Display<T>(T[,] array2d, int n, int m)
        {
            for (var i = 0; i < n; i++)
            {
                for (var j = 0; j < m; j++)
                {
                    Console.Write($"{array2d[i, j]} ");
                }
                Console.WriteLine();
            }
        }
        static public long Gcd(long a, long b)
        {
            if (a == 0 || b == 0) return Max(a, b);
            return a % b == 0 ? b : Gcd(b, a % b);
        }
        static public long LPow(int a, int b) => (long)Pow(a, b);
        static public bool Bit(long x, int dig) => ((1L << dig) & x) != 0;
        static public int Sig(long a) => a == 0 ? 0 : (int)(a / Abs(a));
    }
}