[ICPC2024 Xi'an I] Dumb Robot

You have a dumb robot, and you are going to let it play games with $n$ robots. There is a matrix $A$ with three rows and three columns in the game. We call the number of row $i$ and column $j$ of this matrix $A_{i,j}$. The game goes like this: Two players each choose an integer from $[1,3]$ at the same time. We call the number your robot chooses $i$, and the number the other robot chooses $j$. The score is $A_{i,j}$. In game $i$, your robot will play this game with the $i$ -th robot. The $i$ -th robot has a probability of choosing $1$ as $p_{i,1}$, a probability of choosing $2$ as $p_{i,2}$, and a probability of choosing $3$ as $p_{i,3}$. Your goal is to make the expected value of the score not negative in each game. But your robot is very dumb, so it will choose $1$ with probability $q_1$, $2$ with probability $q_2$, and $3$ with probability $q_3$, and you don't know the value of $q_1,q_2,q_3$. We all know that $q_1+q_2+q_3=1$. If $q_1,q_2,q_3$ are chosen uniformly at random from a set of all possible cases, please calculate the probability that you will reach your goal.

The first line contains one integer $n$($1\le n\le10^4$). Each of the next $3$ lines contains $3$ integers, the $j$ -th integer in the $i$ -th of these lines is $A_{i,j}$($-20\le A_{i,j}\le20$). Each of the next $n$ lines contains $3$ real numbers, the $j$ -th number in the $i$ -th of these lines is $p_{i,j}$. It is guaranteed that $p_{i,1}+p_{i,2}+p_{i,3}=1$ and $0\le p_{i,j}$.

Output the answer to the problem. It is guaranteed that the answer will never be $0$. Your answer is considered correct if its absolute or relative error does not exceed $10^{-2}$. Formally, let your answer be $a$, and the jury's answer be $b$. Your answer is accepted if and only if $\frac{|a-b|}{max(1,|b|)} \leq 10^{-2}$.

1
1 1 1
 -1 2 1
0 -3 2
0.1 0.6 0.3

0.748252

8
1 3 -2
0 0 2
-2 2 1
0.1 0.3 0.6
0 0 1
0.5 0.2 0.3
0 0 1
1 0 0
0 0 1 
0.33 0.33 0.34
0.16 0.16 0.68

0.111111

In example $1$, for example, $(q_1=1,q_2=0,q_3=0)$ is ok. In this case, Your robot will always choose $1$, so no matter what number will robot $1$ choose, the score will always be $1$, which is enough to reach your goal.