Problem:
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へやわり！

N+1 部屋が1列に並んだ新築の建物がある．各部屋はすべて1人用の住居であり，現在どの部屋も空室だが，来月から新たに N 人がここに住む予定である．したがって彼らが暮らし始めると1室は空き部屋になる．

あなたは大家として，彼らの嗜好に合わせた部屋割りを多く提案したいと考えている．ここで部屋割りとは，それぞれの人がどの部屋に住むのかを与える表である．例えば N = 3 のとき，「1人目は4号室，2人目は1号室，3人目は2号室」がありうる一つの部屋割りである．

もちろん，ありうるすべての部屋割りを提案すれば話は早いが，それでは提案の意味がないため，大家の手間と住人の嗜好を考慮していくつか制約を設ける．

まず，提案された部屋割りの一つに彼らが住み始めてから，「別の提案された部屋割りの方に変更したい」と言われるかもしれない．この建物は新築であり，彼らはまだ誰も入居していない新しい部屋が好みであることがわかっているので，単純に誰か1人が空き部屋へ引っ越しをするだけで提案された別の部屋割りに変更できるときに限りこれが起こりうる．しかし大家として，そういう面倒ごとは避けたいため，このような変更が許されないよう，提案する部屋割りをうまく調整したい．
つまり，提案する部屋割りの集合は次を満たさなければならない．
「任意の二つの異なる提案された部屋割り A, B に対して，部屋割り A の空き部屋に誰か1人を移しても部屋割り B にならない」

次に，これから住む N 人にはそれぞれちょうど1人好ましい人がいることがわかっている．したがって，提案するすべての部屋割りは，すべての人に対して好ましい人が隣に住むような部屋割りにする．

これらの条件を満たす部屋割りの集合で，最大の集合の大きさはいくらだろうか？1,000,000,007 で割った余りを求めてほしい．

Input

入力は複数のデータセットからなる．
各データセットは次の形式で表される．

N
a1 a2 ... aN

データセットの 1 行目には，来月建物に住む人数を表す整数 N が与えられる．これは 2 ≤ N ≤ 100,000 を満たす．
2行目には i 番目の人にとって隣の部屋に住んでくれると好ましい人の番号 ai の情報が与えられる．これについては 1 ≤ ai ≤ N かつ ai ≠ i を満たす．
また，データセットの数は 50 を超えない．
入力の終わりは，1個のゼロだけからなる行で表される．

Output

各データセットに対して，問題文にあるような，同時に提案できる部屋割りの数の最大値を，1,000,000,007 で割った余りを1行で出力せよ．

Sample Input

2
2 1
3
3 1 2
10
2 1 1 1 1 1 1 1 1 1
8
2 1 4 3 3 7 6 6
25
2 3 2 5 4 5 8 7 8 11 10 13 12 15 14 17 16 19 18 21 20 23 22 25 24
10
2 1 4 3 6 5 8 7 10 9
0

Output for Sample Input

2
0
0
144
633544712
11520

Solution in C# (judge verdict: Accepted)

using System;
using System.Collections.Generic;
using System.Linq;
using System.IO;
using System.Threading;
using System.Text;
using System.Text.RegularExpressions;
using System.Diagnostics;
using static util;
using P = pair<int, int>;

class Program {
    static void Main(string[] args) {
        var sw = new StreamWriter(Console.OpenStandardOutput()) { AutoFlush = false };
        var solver = new Solver(sw);
        // var t = new Thread(solver.solve, 1 << 26); // 64 MB
        // t.Start();
        // t.Join();
        solver.solve();
        sw.Flush();
    }
}

class Solver {
    StreamWriter sw;
    Scan sc;
    void Prt(string a) => sw.WriteLine(a);
    void Prt<T>(IEnumerable<T> a) => Prt(string.Join(" ", a));
    void Prt(params object[] a) => Prt(string.Join(" ", a));
    public Solver(StreamWriter sw) {
        this.sw = sw;
        this.sc = new Scan();
    }

    public void solve() {
        MyMath.setfacts(200000);
        while (true) {
            int n = sc.Int;
            if (n == 0) return;
            var a = sc.IntArr.Select(x => x - 1).ToArray();
            var rev = new List<int>[n];
            for (int i = 0; i < n; i++)
            {
                rev[i] = new List<int>();
            }
            for (int i = 0; i < n; i++)
            {
                rev[a[i]].Add(i);
            }
            int cnt = 0;
            int c0 = 0;
            var ok = new bool[n];
            for (int i = 0; i < n; i++)
            {
                if (i < a[i] && a[a[i]] == i) {
                    ok[i] = ok[a[i]] = true;
                    ++cnt;
                    if (rev[i].Count == 1 && rev[a[i]].Count == 1) {
                        ++c0;
                        continue;
                    }
                    {
                        int p = i;
                        while (true) {
                            int nex = p;
                            foreach (var item in rev[p])
                            {
                                if (!ok[item]) {
                                    ok[item] = true;
                                    nex = item;
                                    break;
                                }
                            }
                            if (p == nex) break;
                            p = nex;
                        }
                    }
                    {
                        int p = a[i];
                        while (true) {
                            int nex = p;
                            foreach (var item in rev[p])
                            {
                                if (!ok[item]) {
                                    ok[item] = true;
                                    nex = item;
                                    break;
                                }
                            }
                            if (p == nex) break;
                            p = nex;
                        }
                    }
                }
            }
            if (ok.Any(x => !x)) {
                Prt(0);
                continue;
            }
            long ans = solve(cnt + 1, cnt - c0);
            for (int i = 1; i <= c0; i++)
            {
                ans = ans * i * 2 % M;
            }
            for (int i = 1; i <= cnt - c0; i++)
            {
                ans = ans * i * 2 % M;
            }
            Prt(ans);

        }

    }
    long solve(int n, int k) {
        long res = 0;
        for (int i = 0; i < n - k; i++)
        {
            if (i % 2 == 0) {
                res = (res + MyMath.comb(i + k + 1, k + 1)) % M;
            }
            else {
                res = (res + M - MyMath.comb(i + k + 1, k + 1)) % M;
            }
        }
        if ((n + k) % 2 == 1)
            return res * (k + 1) % M;
        else
            return (M - res * (k + 1) % M) % M;


    }
}

class pair<T, U> : IComparable<pair<T, U>> {
    public T v1;
    public U v2;
    public pair() : this(default(T), default(U)) {}
    public pair(T v1, U v2) { this.v1 = v1; this.v2 = v2; }
    public int CompareTo(pair<T, U> a) {
        int c = Comparer<T>.Default.Compare(v1, a.v1);
        return c != 0 ? c : Comparer<U>.Default.Compare(v2, a.v2);
    }
    public override string ToString() => v1 + " " + v2;
    public void Deconstruct(out T a, out U b) { a = v1; b = v2; }
}
static class util {
    public static readonly int M = 1000000007;
    // public static readonly int M = 998244353;
    public static readonly long LM = 1L << 60;
    public static readonly double eps = 1e-11;
    public static void DBG(string a) => Console.Error.WriteLine(a);
    public static void DBG<T>(IEnumerable<T> a) => DBG(string.Join(" ", a));
    public static void DBG(params object[] a) => DBG(string.Join(" ", a));
    public static void Assert(params bool[] conds) {
        if (conds.Any(x => !x)) throw new Exception();
    }
    public static pair<T, U> make_pair<T, U>(T v1, U v2) => new pair<T, U>(v1, v2);
    public static int CompareList<T>(IList<T> a, IList<T> b) where T : IComparable<T> {
        for (int i = 0; i < a.Count && i < b.Count; i++)
            if (a[i].CompareTo(b[i]) != 0) return a[i].CompareTo(b[i]);
        return a.Count.CompareTo(b.Count);
    }
    public static bool inside(int i, int j, int h, int w) => i >= 0 && i < h && j >= 0 && j < w;
    public static readonly int[] dd = { 0, 1, 0, -1 };
    // static readonly string dstring = "RDLU";
    public static IEnumerable<P> adjacents(int i, int j)
        => Enumerable.Range(0, dd.Length).Select(k => new P(i + dd[k], j + dd[k ^ 1]));
    public static IEnumerable<P> adjacents(int i, int j, int h, int w)
        => adjacents(i, j).Where(p => inside(p.v1, p.v2, h, w));
    public static IEnumerable<P> adjacents(this P p) => adjacents(p.v1, p.v2);
    public static IEnumerable<P> adjacents(this P p, int h, int w) => adjacents(p.v1, p.v2, h, w);
    public static IEnumerable<int> all_subset(this int p) {
        for (int i = 0; ; i = i - p & p) {
            yield return i;
            if (i == p) break;
        }
    }
    public static Dictionary<T, int> compress<T>(this IEnumerable<T> a)
        => a.Distinct().OrderBy(v => v).Select((v, i) => new { v, i }).ToDictionary(p => p.v, p => p.i);
    public static Dictionary<T, int> compress<T>(params IEnumerable<T>[] a) => compress(a.SelectMany(x => x));
    public static T[] inv<T>(this Dictionary<T, int> dic) {
        var res = new T[dic.Count];
        foreach (var item in dic) res[item.Value] = item.Key;
        return res;
    }
    public static void swap<T>(ref T a, ref T b) where T : struct { var t = a; a = b; b = t; }
    public static void swap<T>(this IList<T> a, int i, int j) where T : struct { var t = a[i]; a[i] = a[j]; a[j] = t; }
    public static T[] copy<T>(this IList<T> a) {
        var ret = new T[a.Count];
        for (int i = 0; i < a.Count; i++) ret[i] = a[i];
        return ret;
    }
}

class Scan {
    StreamReader sr;
    public Scan() { sr = new StreamReader(Console.OpenStandardInput()); }
    public Scan(string path) { sr = new StreamReader(path); }
    public int Int => int.Parse(Str);
    public long Long => long.Parse(Str);
    public double Double => double.Parse(Str);
    public string Str => sr.ReadLine().Trim();
    public pair<T, U> Pair<T, U>() {
        T a; U b;
        Multi(out a, out b);
        return new pair<T, U>(a, b);
    }
    public P P => Pair<int, int>();
    public int[] IntArr => StrArr.Select(int.Parse).ToArray();
    public long[] LongArr => StrArr.Select(long.Parse).ToArray();
    public double[] DoubleArr => StrArr.Select(double.Parse).ToArray();
    public string[] StrArr => Str.Split(new[]{' '}, StringSplitOptions.RemoveEmptyEntries);
    bool eq<T, U>() => typeof(T).Equals(typeof(U));
    T ct<T, U>(U a) => (T)Convert.ChangeType(a, typeof(T));
    T cv<T>(string s) => eq<T, int>()    ? ct<T, int>(int.Parse(s))
                       : eq<T, long>()   ? ct<T, long>(long.Parse(s))
                       : eq<T, double>() ? ct<T, double>(double.Parse(s))
                       : eq<T, char>()   ? ct<T, char>(s[0])
                                         : ct<T, string>(s);
    public void Multi<T>(out T a) => a = cv<T>(Str);
    public void Multi<T, U>(out T a, out U b) {
        var ar = StrArr; a = cv<T>(ar[0]); b = cv<U>(ar[1]);
    }
    public void Multi<T, U, V>(out T a, out U b, out V c) {
        var ar = StrArr; a = cv<T>(ar[0]); b = cv<U>(ar[1]); c = cv<V>(ar[2]);
    }
    public void Multi<T, U, V, W>(out T a, out U b, out V c, out W d) {
        var ar = StrArr; a = cv<T>(ar[0]); b = cv<U>(ar[1]); c = cv<V>(ar[2]); d = cv<W>(ar[3]);
    }
    public void Multi<T, U, V, W, X>(out T a, out U b, out V c, out W d, out X e) {
        var ar = StrArr; a = cv<T>(ar[0]); b = cv<U>(ar[1]); c = cv<V>(ar[2]); d = cv<W>(ar[3]); e = cv<X>(ar[4]);
    }
}

static class MyMath {
    public static long Mod = util.M;
    // public static long Mod = 1000000007;
    public static bool isprime(long a) {
        if (a < 2) return false;
        for (long i = 2; i * i <= a; i++) if (a % i == 0) return false;
        return true;
    }
    public static bool[] sieve(int n) {
        var p = new bool[n + 1];
        for (int i = 2; i <= n; i++) p[i] = true;
        for (int i = 2; i * i <= n; i++) if (p[i]) for (int j = i * i; j <= n; j += i) p[j] = false;
        return p;
    }
    public static int[] sieve2(int n) {
        var p = new int[n + 1];
        for (int i = 2; i <= n; i++) p[i] = i;
        for (int i = 2; i * i <= n; i++) {
            if (p[i] == i)
                for (int j = i * i; j <= n; j += i)
                    p[j] = Math.Min(p[j], i);
        }
        return p;
    }
    public static bool[] segmentSieve(long l, long r) {
        int sqn = (int)Math.Sqrt(r + 9);
        var ps = getprimes(sqn);
        return segmentSieve(l, r,ps);
    }
    public static bool[] segmentSieve(long l, long r, List<int> ps) {
        var sieve = new bool[r - l + 1];
        for (long i = l; i <= r; i++) sieve[i - l] = true;
        foreach (long p in ps) {
            if (p * p > r) break;
            for (long i = p >= l ? p * p : (l + p - 1) / p * p; i <= r; i += p) sieve[i - l] = false;
        }
        return sieve;
    }
    public static List<int> getprimes(int n) {
        var prs = new List<int>();
        var p = sieve(n);
        for (int i = 2; i <= n; i++) if (p[i]) prs.Add(i);
        return prs;
    }
    public static long pow(long a, long b, long mod) {
        a %= mod;
        if (b < 0) Console.Error.WriteLine($"power number is negative ({a}^{b}).");
        if (b <= 0) return 1;
        var t = pow(a, b / 2, mod);
        if ((b & 1) == 0) return t * t % mod;
        return t * t % mod * a % mod;
    }
    public static long pow(long a, long b) => pow(a, b, Mod);
    public static long inv(long a) => pow(a, Mod - 2);
    public static long gcd(long a, long b) {
        while (b > 0) {
            var t = a % b;
            a = b;
            b = t;
        }
        return a;
    }
    // a x + b y = gcd(a, b)
    public static long extgcd(long a, long b, out long x, out long y) {
        long g = a; x = 1; y = 0;
        if (b > 0) { g = extgcd(b, a % b, out y, out x); y -= a / b * x; }
        return g;
    }

    // return (r, m): x = r (mod. m)
    // return (0, -1) if no answer
    public static pair<long, long> chineserem(IList<long> b, IList<long> m) {
        long r = 0, M = 1;
        for (int i = 0; i < b.Count; ++i) {
            long p, q;
            long d = extgcd(M, m[i], out p, out q); // p is inv of M/d (mod. m[i]/d)
            if ((b[i] - r) % d != 0) return new pair<long, long>(0, -1);
            long tmp = (b[i] - r) / d * p % (m[i]/d);
            r += M * tmp;
            M *= m[i]/d;
        }
        return new pair<long, long>((r % M + M) % M, M);
    }

    // return k: x^k = y (mod. mod) O(sqrt(mod))
    public static long modlog(long x, long y, long mod) {
        if (y == 1) return 0;
        long H = (long)Math.Sqrt(mod) + 1;
        var baby = new Dictionary<long, long>();
        for (long b = 0, xby = y; b < H; b++, xby = (xby * x) % mod) {
            if (!baby.ContainsKey(xby))
                baby.Add(xby, b);
            else
                baby[xby] = b;
        }

        long xH = 1;
        for (int i = 0; i < H; ++i) xH = xH * x % mod;
        for (long a = 1, xaH = xH; a <= H; a++, xaH = (xaH * xH) % mod)
            if (baby.ContainsKey(xaH))
                return a * H - baby[xaH];

        return -1;
    }
    public static long lcm(long a, long b) => a / gcd(a, b) * b;

    static long[] facts, invs;
    public static void setfacts(int n) {
        facts = new long[n + 1];
        facts[0] = 1;
        for (int i = 1; i <= n; i++) facts[i] = facts[i - 1] * i % Mod;
        invs = new long[n + 1];
        invs[n] = inv(facts[n]);
        for (int i = n; i > 0 ; i--) invs[i - 1] = invs[i] * i % Mod;
    }
    public static long fact(long n) {
        if (n < 0) return 0;
        if (facts != null && facts.Length > n) return facts[n];
        long numer = 1;
        for (long i = 1; i <= n; i++) numer = numer * (i % Mod) % Mod;
        return numer;
    }
    public static long perm(long n, long r) {
        if (n < 0 || r < 0 || r > n) return 0;
        if (facts != null && facts.Length > n) return facts[n] * invs[n - r] % Mod;
        long numer = 1;
        for (long i = 0; i < r; i++) numer = numer * ((n - i) % Mod) % Mod;
        return numer;
    }
    public static long comb(long n, long r) {
        if (n < 0 || r < 0 || r > n) return 0;
        if (facts != null && facts.Length > n) return facts[n] * invs[r] % Mod * invs[n - r] % Mod;
        if (n - r < r) r = n - r;
        long numer = 1, denom = 1;
        for (long i = 0; i < r; i++) {
            numer = numer * ((n - i) % Mod) % Mod;
            denom = denom * ((i + 1) % Mod) % Mod;
        }
        return numer * inv(denom) % Mod;
    }
    public static long multi_choose(long n, long r) => comb(n + r - 1, r);
    public static long[][] getcombs(int n) {
        var ret = new long[n + 1][];
        for (int i = 0; i <= n; i++) {
            ret[i] = new long[i + 1];
            ret[i][0] = ret[i][i] = 1;
            for (int j = 1; j < i; j++) ret[i][j] = (ret[i - 1][j - 1] + ret[i - 1][j]) % Mod;
        }
        return ret;
    }
    // nC0, nC2, ..., nCn
    public static long[] getcomb(int n) {
        var ret = new long[n + 1];
        ret[0] = 1;
        for (int i = 0; i < n; i++) ret[i + 1] = ret[i] * (n - i) % Mod * inv(i + 1) % Mod;
        return ret;
    }

    public static class ModMatrix {
        public static long[][] E(int n) {
            var ret = new long[n][];
            for (int i = 0; i < n; i++) { ret[i] = new long[n]; ret[i][i] = 1; }
            return ret;
        }
        public static long[][] pow(long[][] A, long n) {
            if (n == 0) return E(A.Length);
            var t = pow(A, n / 2);
            if ((n & 1) == 0) return mul(t, t);
            return mul(mul(t, t), A);
        }
        public static long dot(long[] x, long[] y) {
            int n = x.Length;
            long ret = 0;
            for (int i = 0; i < n; i++) ret = (ret + x[i] * y[i]) % Mod;
            return ret;
        }
        public static long[][] trans(long[][] A) {
            int n = A[0].Length, m = A.Length;
            var ret = new long[n][];
            for (int i = 0; i < n; i++) { ret[i] = new long[m]; for (int j = 0; j < m; j++) ret[i][j] = A[j][i]; }
            return ret;
        }
        public static long[] mul(long a, long[] x) {
            int n = x.Length;
            var ret = new long[n];
            for (int i = 0; i < n; i++) ret[i] = a * x[i] % Mod;
            return ret;
        }
        public static long[] mul(long[][] A, long[] x) {
            int n = A.Length;
            var ret = new long[n];
            for (int i = 0; i < n; i++) ret[i] = dot(x, A[i]);
            return ret;
        }
        public static long[][] mul(long a, long[][] A) {
            int n = A.Length;
            var ret = new long[n][];
            for (int i = 0; i < n; i++) ret[i] = mul(a, A[i]);
            return ret;
        }
        public static long[][] mul(long[][] A, long[][] B) {
            int n = A.Length;
            var Bt = trans(B);
            var ret = new long[n][];
            for (int i = 0; i < n; i++) ret[i] = mul(Bt, A[i]);
            return ret;
        }
    }
}