Running in the Plane

You are given a set $S=\\{(a\_1,b\_1) ,(a\_2,b\_2) ,\dots ,(a\_n,b\_n)\\}$ of $n$ points in the plane. All coordinates of $S$ are integers.

A set $T=\\{(c\_1,d\_1) ,(c\_2,d\_2) ,\dots ,(c\_m,d\_m)\\}$ of $m$ 2-dimensional vectors is called a good set of $S$ if it satisfies the following:

1. There exists a nonempty finite sequence $((x\_0,y\_0) ,(x\_1,y\_1) ,\dots ,(x\_l,y\_l))$ of points in the plane such that

   1. $(x\_0,y\_0) =(0,0)$.
   2. For all points $p$ in $S$, there exists an integer $i$ ($0\le i\le l$) such that $(x\_i,y\_i) =p$.
   3. For all integers $i$ ($0\le i\<l$), the vector $(x\_{i+1}-x\_i,y\_{i+1}-y\_i)$ is in $T$.

2. For all integers $i$ ($1\le i\le m$), two numbers $c\_i$ and $d\_i$ are integers between $-10^{18}$ and $10^{18}$ inclusive.

Find any good set of minimum size.

The input consists of multiple test cases. The first line contains an integer $Q$ — the number of test cases. The description of the test cases follows. For each test case:

• The first line of the test case contains an integer $n$ — the number of points in $S$.
• The $i$-th of the next $n$ lines contains two integers $a\_i$ and $b\_i$ — the coordinates of each point in $S$.

For each test case:

• Let $T=\\{(c\_1,d\_1) ,(c\_2,d\_2) ,\dots ,(c\_m,d\_m)\\}$ be a minimum-size good set of $S$.
• In the first line of the test case, print an integer $m$ — the number of vectors in $T$.
• In the $i$-th of the next $m$ lines, print two integers $c\_i$ and $d\_i$ — the coordinates of each vector.

If there are multiple solutions, print any of them.

It can be proved that, under the constraints of this problem, a good set of $S$ with size at most $10\times n$ always exists.

• $1\le Q\le 50\\, 000$
• The sum of $n$ over all test cases does not exceed $10^5$.
• $2\le n\le 10^5$
• $-10^8\le a\_i,b\_i\le 10^8$ ($1\le i\le n$)
• $(a\_i,b\_i)\ne(a\_j,b\_j)$ ($1\le i\<j\le n$)
• $m\ge 0$
• $-10^{18}\le c\_i,d\_i\le 10^{18}$ ($1\le i\le m$)
• $(c\_i,d\_i)\ne(c\_j,d\_j)$ ($1\le i\<j\le m$)

In the first test case, $T=\\{(-10,10)\\}$ is a minimum-size good set of $S=\\{(-30,30) ,(-50,50)\\}$.

We can take a sequence $((0,0) ,(-10,10) ,(-20,20) ,\underline{(-30,30)} ,(-40,40) ,\underline{(-50,50)})$. Here, the underlined points are in $S$.

In the second test case, $T=\\{(1,0) ,(1,1)\\}$ is a minimum-size good set of $S=\\{(2,1) ,(1,0) ,(4,1)\\}$.

We can take a sequence $((0,0) ,\underline{(1,0)} ,\underline{(2,1)} ,(3,1) ,\underline{(4,1)})$.