Exercising Walk
Contest: Codeforces Round 630 (Div. 2)
Difficulty rating: 1100
Tags: greedy, implementation, math

Alice has a cute cat. To keep her cat fit, Alice wants to design an exercising walk for her cat!

Initially, Alice's cat is located in a cell $$$(x,y)$$$ of an infinite grid. According to Alice's theory, cat needs to move:

- exactly $$$a$$$ steps left: from $$$(u,v)$$$ to $$$(u-1,v)$$$;
- exactly $$$b$$$ steps right: from $$$(u,v)$$$ to $$$(u+1,v)$$$;
- exactly $$$c$$$ steps down: from $$$(u,v)$$$ to $$$(u,v-1)$$$;
- exactly $$$d$$$ steps up: from $$$(u,v)$$$ to $$$(u,v+1)$$$.

Note that the moves can be performed in an arbitrary order. For example, if the cat has to move $$$1$$$ step left, $$$3$$$ steps right and $$$2$$$ steps down, then the walk right, down, left, right, right, down is valid.

Alice, however, is worrying that her cat might get lost if it moves far away from her. So she hopes that her cat is always in the area $$$[x_1,x_2]\times [y_1,y_2]$$$, i.e. for every cat's position $$$(u,v)$$$ of a walk $$$x_1 \le u \le x_2$$$ and $$$y_1 \le v \le y_2$$$ holds.

Also, note that the cat can visit the same cell multiple times.

Can you help Alice find out if there exists a walk satisfying her wishes?

Formally, the walk should contain exactly $$$a+b+c+d$$$ unit moves ($$$a$$$ to the left, $$$b$$$ to the right, $$$c$$$ to the down, $$$d$$$ to the up). Alice can do the moves in any order. Her current position $$$(u, v)$$$ should always satisfy the constraints: $$$x_1 \le u \le x_2$$$, $$$y_1 \le v \le y_2$$$. The staring point is $$$(x, y)$$$.

You are required to answer $$$t$$$ test cases independently.

Input format:
The first line contains a single integer $$$t$$$ ($$$1 \le t \le 10^3$$$) — the number of testcases.

The first line of each test case contains four integers $$$a$$$, $$$b$$$, $$$c$$$, $$$d$$$ ($$$0 \le a,b,c,d \le 10^8$$$, $$$a+b+c+d \ge 1$$$).

The second line of the test case contains six integers $$$x$$$, $$$y$$$, $$$x_1$$$, $$$y_1$$$, $$$x_2$$$, $$$y_2$$$ ($$$-10^8 \le x_1\le x \le x_2 \le 10^8$$$, $$$-10^8 \le y_1 \le y \le y_2 \le 10^8$$$).

Output format:
For each test case, output "YES" in a separate line, if there exists a walk satisfying her wishes. Otherwise, output "NO" in a separate line.

You can print each letter in any case (upper or lower).

Constraints:
Time limit: 2.0 s
Memory limit: 512.0 MB

Note:
In the first test case, one valid exercising walk is $$$$$$(0,0)\rightarrow (-1,0) \rightarrow (-2,0)\rightarrow (-2,1) \rightarrow (-2,2)\rightarrow (-1,2)\rightarrow(0,2)\rightarrow (0,1)\rightarrow (0,0) \rightarrow (-1,0)$$$$$$

Example input:
6
3 2 2 2
0 0 -2 -2 2 2
3 1 4 1
0 0 -1 -1 1 1
1 1 1 1
1 1 1 1 1 1
0 0 0 1
0 0 0 0 0 1
5 1 1 1
0 0 -100 -100 0 100
1 1 5 1
0 0 -100 -100 100 0

Example output:
Yes
No
No
Yes
Yes
Yes

Editorial:
The key observation is x-axis and y-axis is independent in this task as the area is a rectangle. Therefore, we should only consider 1D case (x-axis, for example). The optimal path to choose alternates between right and left moves until only one type of move is possible. And sometimes there is no place to make even one move, which has to handled separately. So the verdict is "Yes" if and only if $$$x_1 \le x-a+b \le x_2$$$ and ($$$x_1<x_2$$$ or $$$a+b=0$$$).