human
Can you provide a Python code to print all integers that you can write on a card so that the described condition fulfilled given the following:
Everybody knows what an arithmetic progression is. Let us remind you just in case that an arithmetic progression is such sequence of numbers a1, a2, ..., an of length n, that the following condition fulfills: 

a2 - a1 = a3 - a2 = a4 - a3 = ... = ai + 1 - ai = ... = an - an - 1.

For example, sequences [1, 5], [10], [5, 4, 3] are arithmetic progressions and sequences [1, 3, 2], [1, 2, 4] are not.

Alexander has n cards containing integers. Arthur wants to give Alexander exactly one more card with a number so that he could use the resulting n + 1 cards to make an arithmetic progression (Alexander has to use all of his cards).

Arthur has already bought a card but he hasn't written a number on it. Help him, print all integers that you can write on a card so that the described condition fulfilled.

Input

The first line contains integer n (1 ≤ n ≤ 105) — the number of cards. The next line contains the sequence of integers — the numbers on Alexander's cards. The numbers are positive integers, each of them doesn't exceed 108.

Output

If Arthur can write infinitely many distinct integers on the card, print on a single line -1.

Otherwise, print on the first line the number of integers that suit you. In the second line, print the numbers in the increasing order. Note that the numbers in the answer can exceed 108 or even be negative (see test samples).

Examples

Input

3
4 1 7

Output

2
-2 10

Input

1
10

Output

-1

Input

4
1 3 5 9

Output

1
7

Input

4
4 3 4 5

Output

0

Input

2
2 4

Output

3
0 3 6
Using python3 can you solve the prior task?
assistant
#Here is Python code to print all integers that you can write on a card so that the described condition fulfilled.

def main(inp):
    n = int(input())
    arr = split_inp_int(inp)
    arr.sort()

    if n == 1:
        print(-1)
        return
    if n == 2:
        a, b = arr
        diff = b-a
        if diff == 0:
            print(1)
            print(a)
        elif diff % 2 == 0:
            print(3)
            print(a-diff, a+diff//2,  b+diff)
        else:
            print(2)
            print(a-diff, b+diff)
        return

    diffs = [arr[i]-arr[i-1] for i in range(1, n)]
    if len(set(diffs)) > 2:
        print(0)
        return
    if len(set(diffs)) == 1:
        diff = arr[1]-arr[0]
        if diff == 0:
            print(1)
            print(arr[0])
        else:
            print(2)
            print(arr[0]-diff, arr[-1]+diff)
        return

    diffs_counter = [(k, v) for k, v in Counter(diffs).items()]
    # print(diffs_counter)
    diffs_counter.sort()
    if diffs_counter[1][1] != 1:
        print(0)
        return
    common_diff = diffs_counter[0][0]
    other_diff = diffs_counter[1][0]
    if other_diff != common_diff*2:
        print(0)
        return
    print(1)
    print(arr[0] + common_diff*diffs.index(other_diff) + common_diff)

def split_inp_int(inp):
    return list(map(int, inp().split()))

def use_fast_io():
    import sys

    class InputStorage:
        def __init__(self, lines):
            lines.reverse()
            self.lines = lines

        def input_func(self):
            if self.lines:
                return self.lines.pop()
            else:
                return ""

    return input
    input_storage_obj = InputStorage(sys.stdin.readlines())
    return input_storage_obj

from collections import Counter, defaultdict
from functools import reduce
import operator
import math

def product(arr_):
    return reduce(operator.mul, arr_, 1)

if __name__ == "__main__":

    main(use_fast_io())