Functions and Minimum Spanning Trees

A simple connected graph with $V$ vertices numbered from $1$ to $V$ and $E$ edges is given.

The weight of each edge is a quadratic function $at^2+b_it+c_i$ that changes with time $t$. The coefficient $a$ is the same for every edge.

Define $f(t)$ as the sum of the weights of the minimum spanning tree at time $t$.

Given integers $t_1$ and $t_2$, find the value of $\int_{t_1}^{t_2}f(t)dt$.

A minimum spanning tree is a tree, among the subgraphs that connect all vertices of the given graph, whose total weight is minimal.

The first line contains integers $V$, $E$, and $a$, separated by spaces. ($1 \le V \le 100$, $1 \le E \le 250$, $-1\,000 \le a \le 1\,000$, $a \neq 0$)

Each of the next $E$ lines contains four integers $X$, $Y$, $b_i$, and $c_i$, separated by spaces. ($1 \le X, Y \le V$, $-1\,000 \le b_i, c_i \le 1\,000$)

This means the weight of the edge connecting vertex $X$ and vertex $Y$ is $at^2+b_it+c_i$.

The last line contains integers $t_1$ and $t_2$, separated by a space. ($-1\,000 \le t_1 \le t_2 \le 1\,000$)

Print $m \times n^{-1} \bmod (10^9+7)$, where the value of the integral is the reduced fraction $\frac{m}{n}$ with integer $m$ and positive integer $n$. $n^{-1}$ is the multiplicative inverse of $n$ modulo $10^9+7$.

It can be proven that the answer can be written in this form for every $n$ that is not a multiple of $10^9+7$.

If needed, you may use $(ab^{-1})+(cd^{-1}) \equiv (ad+bc)\times(bd)^{-1} \pmod {10^9+7}$.