Euclid Guess
Description
Let's consider Euclid's algorithm for finding the greatest common divisor, where $t$ is a list:
function Euclid(a, b):
if a < b:
swap(a, b)
if b == 0:
return a
r = reminder from dividing a by b
if r > 0:
append r to the back of t
return Euclid(b, r)
There is an array $p$ of pairs of positive integers that are not greater than $m$. Initially, the list $t$ is empty. Then the function is run on each pair in $p$. After that the list $t$ is shuffled and given to you.
You have to find an array $p$ of any size not greater than $2 \cdot 10^4$ that produces the given list $t$, or tell that no such array exists.
The first line contains two integers $n$ and $m$ ($1 \le n \le 10^3$, $1 \le m \le 10^9$) — the length of the array $t$ and the constraint for integers in pairs.
The second line contains $n$ integers $t_1, t_2, \ldots, t_n$ ($1 \le t_i \le m$) — the elements of the array $t$.
- If the answer does not exist, output $-1$.
- If the answer exists, in the first line output $k$ ($1 \le k \le 2 \cdot 10^4$) — the size of your array $p$, i. e. the number of pairs in the answer.
The $i$-th of the next $k$ lines should contain two integers $a_i$ and $b_i$ ($1 \le a_i, b_i \le m$) — the $i$-th pair in $p$.
If there are multiple valid answers you can output any of them.
Input
The first line contains two integers $n$ and $m$ ($1 \le n \le 10^3$, $1 \le m \le 10^9$) — the length of the array $t$ and the constraint for integers in pairs.
The second line contains $n$ integers $t_1, t_2, \ldots, t_n$ ($1 \le t_i \le m$) — the elements of the array $t$.
Output
- If the answer does not exist, output $-1$.
- If the answer exists, in the first line output $k$ ($1 \le k \le 2 \cdot 10^4$) — the size of your array $p$, i. e. the number of pairs in the answer.
The $i$-th of the next $k$ lines should contain two integers $a_i$ and $b_i$ ($1 \le a_i, b_i \le m$) — the $i$-th pair in $p$.
If there are multiple valid answers you can output any of them.
Samples
7 20
1 8 1 6 3 2 3
3
19 11
15 9
3 7
2 10
7 1
-1
2 15
1 7
1
15 8
1 1000000000
845063470
-1
Note
In the first sample let's consider the array $t$ for each pair:
- $(19,\, 11)$: $t = [8, 3, 2, 1]$;
- $(15,\, 9)$: $t = [6, 3]$;
- $(3,\, 7)$: $t = [1]$.
So in total $t = [8, 3, 2, 1, 6, 3, 1]$, which is the same as the input $t$ (up to a permutation).
In the second test case it is impossible to find such array $p$ of pairs that all integers are not greater than $10$ and $t = [7, 1]$
In the third test case for the pair $(15,\, 8)$ array $t$ will be $[7, 1]$.