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Ex - General General

Score : $600$ points

Problem Statement

Solve the following problem for $T$ test cases.

A piece is placed at the origin $(0, 0)$ on an $xy$-plane. You may perform the following operation any number of (possibly zero) times:

• Choose an integer $i$ such that $1 \leq i \leq 8$ and $s_i=$ 1. Let $(x, y)$ be the current coordinates where the piece is placed.
  • If $i=1$, move the piece to $(x+1,y)$.
  • If $i=2$, move the piece to $(x+1,y+1)$.
  • If $i=3$, move the piece to $(x,y+1)$.
  • If $i=4$, move the piece to $(x-1,y+1)$.
  • If $i=5$, move the piece to $(x-1,y)$.
  • If $i=6$, move the piece to $(x-1,y-1)$.
  • If $i=7$, move the piece to $(x,y-1)$.
  • If $i=8$, move the piece to $(x+1,y-1)$.

Your objective is to move the piece to $(A, B)$.
Find the minimum number of operations needed to achieve the objective. If it is impossible, print -1 instead.

Constraints

• $1 \leq T \leq 10^4$
• $-10^9 \leq A,B \leq 10^9$
• $s_i$ is 0 or 1.
• $T$, $A$, and $B$ are integers.

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Input

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

$T$

$\mathrm{case}_1$

$\mathrm{case}_2$

$\vdots$

$\mathrm{case}_T$

Here, $\mathrm{case}_i$ denotes the $i$-th test case.

Each test case is given in the following format:

``` $A$ $B$ $s_1 s_2 s_3 s_4 s_5 s_6 s_7 s_8$ ```

Output

Print $T$ lines in total.
The $i$-th line should contain the answer to the $i$-th test case.

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7
5 3 10101010
5 3 01010101
5 3 11111111
5 3 00000000
0 0 11111111
0 1 10001111
-1000000000 1000000000 10010011

8
5
5
-1
0
-1
1000000000