Let's consider Euclid's algorithm for finding thegreatest 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.
7 201 8 1 6 3 2 3
319 1115 93 7
2 107 1
-1
2 151 7
115 8
1 1000000000845063470
-1
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 $$$[7, 1]$$$.
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