P6211 "SWTR-4" Meeting in the Forest

Whenever the moon is full, the tribe cats from five clans gather on a small island to hold the monthly forest meeting. In order to learn the characteristics of the other five clans of cats, Juxing disguised themself as a lone cat and came to the island...

At the forest meeting there are $n$ cats, and the strength value of each cat is $a_i$. So Juxing wrote down the following equation: $$x^n+\sum_{i=1}^{n}a_ix^{n-i}=0$$ - In plain words, this equation is $x^n+a_1x^{n-1}+a_2x^{n-2}+\cdots+a_{n-1}x+a_n=0$. With their excellent (cu) math skills, Juxing knows that this equation has $n$ roots in the complex numbers. Let these $n$ roots be $x_1, x_2, ..., x_n$. Next, Juxing wants to know how strong the cats at the forest meeting are, so they wrote the following expression: $$\sum_{i=1}^{n}(b_i\times \sum_{1\le j_1 < j_2

The first line contains an integer $n$ — the degree of the equation. The second line contains $n$ integers $a_1, a_2, ... a_n$ — as described in the statement. The third line contains $n$ integers $b_1, b_2, ... b_n$ — same as above.

Output one line containing one number, the value of this expression modulo $10^9+7$.

Sample 1 Explanation: The original equation is $x^2-2x+1=0$, and $x_1=x_2=1$. The value of the expression is $x_1+x_2+x_1x_2=1+1+1=3$. Sample 2 Explanation: The original equation is $x^3-3x^2+4=0$, and $x_1=-1,x_2=x_3=2$. The value of the expression is $$\begin{aligned}&2\cdot (x_1+x_2+x_3)+3\cdot(x_1x_2+x_1x_3+x_2x_3)+4\cdot x_1x_2x_3\\=\ &2\times(-1+2+2)+3\times(-2+(-2)+4)+4\times (-4)\\=\ &-10\end{aligned}$$ Since $-10$ is negative, output $-10+(10^9+7)=999999997$. Constraints: For $10\%$ of the testdata, $n=1$. For another $20\%$ of the testdata, $n=2$. For $40\%$ of the testdata, $n\leq 10$. For $60\%$ of the testdata, $n\leq 10^3$. For $100\%$ of the testdata, $1\leq n \le 2 \times 10^5$, $-10^9 \le a_i, b_i \le 10^9$. Tips: [Vieta's formulas](https://baike.baidu.com/item/%E9%9F%A6%E8%BE%BE%E5%AE%9A%E7%90%86/105027?fr=aladdin) may be helpful. Source: [Sweet Round 04](https://www.luogu.com.cn/contest/26414)$\ \ $C idea & std: [蒟蒻的名字](https://www.luogu.com.cn/user/147999) Translated by ChatGPT 5