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F - Teleporter Takahashi

Score : $500$ points

Problem Statement

Takahashi is on an $xy$-plane. Initially, he is at point $(s _ x,s _ y)$, and he wants to reach point $(t _ x,t _ y)$.

On the $xy$-plane is a rectangle $R\coloneqq\lbrace(x,y)\mid a-0.5\leq x\leq b+0.5,c-0.5\leq y\leq d+0.5\rbrace$. Consider the following operation:

• Choose a lattice point $(x,y)$ contained in the rectangle $R$. Takahashi teleports to the point symmetric to his current position with respect to point $(x,y)$.

Determine if he can reach point $(t _ x,t _ y)$ after repeating the operation above between $0$ and $10^6$ times, inclusive. If it is possible, construct a sequence of operations that leads him to point $(t _ x,t _ y)$.

Constraints

• $0\leq s _ x,s _ y,t _ x,t _ y\leq2\times10^5$
• $0\leq a\leq b\leq2\times10^5$
• $0\leq c\leq d\leq2\times10^5$
• All values in the input are integers.

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Input

The input is given from Standard Input in the following format:

$s _ x$ $s _ y$

$t _ x$ $t _ y$

$a$ $b$ $c$ $d$

Output

In the first line, print Yes if Takahashi can reach point $(t _ x,t _ y)$ after repeating the operation between $0$ and $10^6$ times, inclusive, and No otherwise. If and only if you print Yes in the first line, print $d$ more lines, where $d$ is the length of the sequence of operations you have constructed ($d$ must satisfy $0\leq d\leq10^6$). The $(1+i)$-th line $(1\leq i\leq d)$ should contain the space-separated coordinates of the point $(x, y)\in R$, in this order, that is chosen in the $i$-th operation.

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1 2
7 8
7 9 0 3

Yes
7 0
9 3
7 1
8 1

For example, the following choices lead him from $(1,2)$ to $(7,8)$.

• Choose $(7,0)$. Takahashi moves to $(13,-2)$.
• Choose $(9,3)$. Takahashi moves to $(5,8)$.
• Choose $(7,1)$. Takahashi moves to $(9,-6)$.
• Choose $(8,1)$. Takahashi moves to $(7,8)$.

Any output that satisfies the conditions is accepted; for example, printing

``` Yes 7 3 9 0 7 2 9 1 8 1 ```

is also accepted.

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0 0
8 4
5 5 0 0

No

No sequence of operations leads him to point $(8,4)$.

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1 4
1 4
100 200 300 400

Yes

Takahashi may already be at the destination in the beginning.

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22 2
16 7
14 30 11 14

No