Diagonal slices of a rectangle union

Vera has $N$ rectangles whose sides are parallel to the axes. The $i$-th rectangle has opposite corners $(a_i, b_i)$ and $(c_i, d_i)$. Let $U$ be the union of the $N$ rectangles.

For a real number $s$, the intersection of $U$ with the line $y = s - x$ is a set of disjoint segments, and some of them may shrink to a single point. Let $f(s)$ be the total length of those segments, and let $f(s) = 0$ when the intersection is empty.

You are given integers $L$ and $R$. Let $S = \sum_{s=L}^{R} f(s)$, where $s$ runs over the integers from $L$ to $R$. This sum is always $V\sqrt{2}$ for some integer $V$. Find $V$.

The first line contains three integers $N$, $L$, $R$ ($1 \le N \le 10^3$, $-2 \times 10^8 \le L < R \le 2 \times 10^8$).

Each of the next $N$ lines contains four integers $a_i$, $b_i$, $c_i$, $d_i$ ($-10^8 \le a_i < c_i \le 10^8$, $-10^8 \le b_i < d_i \le 10^8$).

Print $V$ on one line.

The figure below shows the first example at $s = 0$. There $f(0)$ is the total length of the two thick blue segments. For that example $S = 7\sqrt{2} \approx 9.899$.