Problem:
Score : 500 points

Problem Statement

An adult game master and N children are playing a game on an ice rink.
The game consists of K rounds.
In the i-th round, the game master announces:

Form groups consisting of A_i children each!

Then the children who are still in the game form as many groups of A_i children as possible.
One child may belong to at most one group.
Those who are left without a group leave the game. The others proceed to the next round.
Note that it's possible that nobody leaves the game in some round.

In the end, after the K-th round, there are exactly two children left, and they are declared the winners.

You have heard the values of A_1, A_2, ..., A_K. You don't know N, but you want to estimate it.

Find the smallest and the largest possible number of children in the game before the start, or determine that no valid values of N exist.

Constraints

1 \leq K \leq 10^5

2 \leq A_i \leq 10^9

All input values are integers.

Input

Input is given from Standard Input in the following format:

K
A_1 A_2 ... A_K

Output

Print two integers representing the smallest and the largest possible value of N, respectively,
or a single integer -1 if the described situation is impossible.

Sample Input 1

4
3 4 3 2

Sample Output 1

6 8

For example, if the game starts with 6 children, then it proceeds as follows:

In the first round, 6 children form 2 groups of 3 children, and nobody leaves the game.

In the second round, 6 children form 1 group of 4 children, and 2 children leave the game.

In the third round, 4 children form 1 group of 3 children, and 1 child leaves the game.

In the fourth round, 3 children form 1 group of 2 children, and 1 child leaves the game.

The last 2 children are declared the winners.

Sample Input 2

5
3 4 100 3 2

Sample Output 2

-1

This situation is impossible.
In particular, if the game starts with less than 100 children, everyone leaves after the third round.

Sample Input 3

10
2 2 2 2 2 2 2 2 2 2

Sample Output 3

2 3

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

using System;
using System.Collections;
using System.Collections.Generic;
using System.Diagnostics;
using System.IO;
using System.Linq;
using System.Numerics;
using System.Runtime.CompilerServices;
using System.Text;
using static System.Math;
using static Extensions;
using static MathExtensions;

public static class Program
{
    [DebuggerDisplay("({Min}, {Max}) ({Value})")]
    public struct State
    {
        public long Min, Max, Value;

        public State(long min, long max)
        {
            Assert(min <= max);
            Min = min;
            Max = max;
            Value = -1;
        }

        public State(long value)
        {
            Min = -1;
            Max = -1;
            Value = value;
        }
    }

    public static void Solve()
    {
        var k = I;
        var a = Repeat(k, () => L).ToArray();

        if (a[k - 1] != 2)
        {
            Console.WriteLine(-1);
            Exit(0);
        }

        var n = new State[k + 1];
        n[k] = new State(2, 2);

        for (var i = k - 1; i >= 0; i--)
        {
            if (a[i] > n[i + 1].Max)
            {
                Console.WriteLine(-1);
                Exit(0);
            }

            n[i + 1] = new State(n[i + 1].Max / a[i] * a[i]);
            n[i] = new State(n[i + 1].Value, n[i + 1].Value + a[i] - 1);
        }

        var min = n[0].Min;
        var max = n[0].Max;
        Array.Clear(n, 0, n.Length);
        n[k] = new State(2, 2);

        for (var i = k - 1; i >= 0; i--)
        {
            if (n[i + 1].Value == -1)
            {
                var tmp = DivCeil(n[i + 1].Min, a[i]) * a[i];
                if (tmp > n[i + 1].Max)
                {
                    if ((n[i + 2].Value += a[i + 1]) > n[i + 1].Max)
                    {
                        Console.WriteLine($"{min} {max}");
                        Exit(0);
                    }
                    i += 2;
                    continue;
                }
                n[i + 1].Value = tmp;
            }
            n[i] = new State(n[i + 1].Value, n[i + 1].Value + a[i] - 1);
        }

        min = n[0].Min;
        Console.WriteLine($"{min} {max}");
    }

    #region Scanners
    static TextScanner _ts;
    static char C => char.Parse(_ts.Next());
    static string S => _ts.Next();
    static int I => int.Parse(_ts.Next());
    static long L => long.Parse(_ts.Next());
    static BigInteger B => BigInteger.Parse(_ts.Next());
    static double D => double.Parse(_ts.Next());
    static decimal M => decimal.Parse(_ts.Next());
    #endregion

    public static void Main()
    {
        var sw = new StreamWriter(Console.OpenStandardOutput());
        sw.NewLine = "\n";
#if DEBUG
        sw.AutoFlush = true;
#else
        sw.AutoFlush = false;
#endif
        Console.SetOut(sw);
        _ts = new TextScanner(Console.In);
        Solve();
        Console.Out.Flush();
    }
}

public static partial class Extensions
{
}

#region Library
public class TextScanner
{
    private readonly TextReader _tr;

    public TextScanner(TextReader tr)
    {
        _tr = tr;
    }

    public string Next()
    {
        var sb = new StringBuilder();
        int i;
        do
        {
            i = _tr.Read();
            if (i == -1) throw new EndOfStreamException();
        }
        while (char.IsWhiteSpace((char)i));
        while (i != -1 && !char.IsWhiteSpace((char)i))
        {
            sb.Append((char)i);
            i = _tr.Read();
        }
        return sb.ToString();
    }
}

[DebuggerStepThrough]
public static partial class Extensions
{
    [MethodImpl(MethodImplOptions.AggressiveInlining)]
    public static void Assert(bool condition)
    {
        if (!condition) throw new Exception("Assertion failed");
    }

    public static string AsString(this IEnumerable<char> source) => new string(source.ToArray());

    public static Dictionary<T, int> Bucket<T>(this IEnumerable<T> source) where T : IEquatable<T>
    {
        var dict = new Dictionary<T, int>();
        foreach (var item in source) if (dict.ContainsKey(item)) dict[item]++; else dict[item] = 1;
        return dict;
    }

    public static int[] Bucket<T>(this IEnumerable<T> source, int maxValue, Func<T, int> selector)
    {
        var arr = new int[maxValue + 1];
        foreach (var item in source) arr[selector(item)]++;
        return arr;
    }

    public static IEnumerable<int> CumSum(this IEnumerable<int> source)
    {
        var sum = 0;
        foreach (var item in source) yield return sum += item;
    }

    public static IEnumerable<long> CumSum(this IEnumerable<long> source)
    {
        var sum = 0L;
        foreach (var item in source) yield return sum += item;
    }

    public static void Exit(int exitCode)
    {
        Console.Out.Flush();
        Environment.Exit(exitCode);
    }

    [MethodImpl(MethodImplOptions.AggressiveInlining)]
    public static void ForEach<T>(this IEnumerable<T> source, Action<T> action)
    {
        foreach (var item in source) action(item);
    }

    [MethodImpl(MethodImplOptions.AggressiveInlining)]
    public static void ForEach<T, _>(this IEnumerable<T> source, Func<T, _> func)
    {
        foreach (var item in source) func(item);
    }

    [MethodImpl(MethodImplOptions.AggressiveInlining)]
    public static void ForEach<T>(this IEnumerable<T> source, Action<T, int> action)
    {
        var i = 0;
        foreach (var item in source) action(item, i++);
    }

    [MethodImpl(MethodImplOptions.AggressiveInlining)]
    public static void ForEach<T, _>(this IEnumerable<T> source, Func<T, int, _> func)
    {
        var i = 0;
        foreach (var item in source) func(item, i++);
    }

    [MethodImpl(MethodImplOptions.AggressiveInlining)]
    public static void Repeat(int count, Action action)
    {
        for (var i = 0; i < count; i++) action();
    }

    [MethodImpl(MethodImplOptions.AggressiveInlining)]
    public static void Repeat(int count, Action<int> action)
    {
        for (var i = 0; i < count; i++) action(i);
    }

    public static IEnumerable<T> Repeat<T>(int count, Func<T> func)
    {
        for (var i = 0; i < count; i++) yield return func();
    }

    public static IEnumerable<T> Repeat<T>(int count, Func<int, T> func)
    {
        for (var i = 0; i < count; i++) yield return func(i);
    }

    public static void Swap<T>(ref T x, ref T y)
    {
        var tmp = x; x = y; y = tmp;
    }
}

[DebuggerStepThrough]
public static class MathExtensions
{
    public static int DivCeil(int left, int right)
        => left / right + (left % right == 0 ? 0 : 1);

    public static long DivCeil(long left, long right)
        => left / right + (left % right == 0L ? 0L : 1L);

    public static int Gcd(int left, int right)
    {
        int r;
        while ((r = left % right) != 0) { left = right; right = r; }
        return right;
    }

    public static long Gcd(long left, long right)
    {
        long r;
        while ((r = left % right) != 0L) { left = right; right = r; }
        return right;
    }

    public static BigInteger Gcd(BigInteger left, BigInteger right)
        => BigInteger.GreatestCommonDivisor(left, right);

    public static int HighestOneBit(int x)
    {
        x |= x >> 01;
        x |= x >> 02;
        x |= x >> 04;
        x |= x >> 08;
        x |= x >> 16;
        return x - (x >> 1);
    }

    public static long HighestOneBit(long x)
    {
        x |= x >> 01;
        x |= x >> 02;
        x |= x >> 04;
        x |= x >> 08;
        x |= x >> 16;
        x |= x >> 32;
        return x - (x >> 1);
    }

    public static int Lcm(int left, int right) => left / Gcd(left, right) * right;

    public static long Lcm(long left, long right) => left / Gcd(left, right) * right;

    public static BigInteger Lcm(BigInteger left, BigInteger right)
        => left / Gcd(left, right) * right;

    public static int Pow(int value, int exponent)
    {
        var r = 1;
        while (exponent > 0)
        {
            if ((exponent & 1) == 1) r *= value;
            value *= value;
            exponent >>= 1;
        }
        return r;
    }

    public static long Pow(long value, int exponent)
    {
        var r = 1L;
        while (exponent > 0)
        {
            if ((exponent & 1) == 1) r *= value;
            value *= value;
            exponent >>= 1;
        }
        return r;
    }

    public static long Fact(int value)
    {
        var r = 1L;
        for (var i = 2; i <= value; i++) r *= i;
        return r;
    }
}
#endregion