Segment Drawing

You are given some points, each paired with a line segment. All points lie strictly above the x-axis, and each line segment lies on the x-axis. No two line segments share a point.

For each point and segment pair, draw one new segment that connects the point to its segment. The pairing is fixed, so you cannot change which point connects to which segment.

No two drawn segments may strictly intersect. Two drawn segments may touch at a shared endpoint.

Find the minimum total length of all drawn segments, or determine that no valid drawing exists.

The first line contains a single integer $n$ ($1 \le n \le 10^5$), the number of point and segment pairs.

Each of the next $n$ lines contains four integers $x$, $y$, $l$, $r$ ($-10^6 \le x \le 10^6$, $0 < y \le 10^6$, $-10^6 \le l \le r \le 10^6$). The line describes a point at $(x, y)$ and its segment with endpoints $(l, 0)$ and $(r, 0)$.

The segments are given from left to right. No two segments share a common point. Two points may be at the same location.

Output a single real number, the minimum total length of all drawn segments. An answer is accepted if its absolute or relative error is at most $10^{-6}$. If the segments cannot be drawn without a strict intersection, output $-1$.