# Find a Horizontal Cut That Bisects Rectangle Area Several non-overlapping rectangular cakes lie on a plane. A cake is represented by `[x, y, width, height]`, where `(x, y)` is its lower-left corner. Find the height of a horizontal line `y = k` such that the total cake area strictly below the line equals the total cake area strictly above it. Cake material crossed by the line is split between the two sides. ```python def horizontal_bisector(rectangles: list[list[float]]) -> float: ... ``` The input guarantees a unique answer. Return `k`; an absolute or relative error of at most `1e-6` is accepted. The rectangles can be very far apart, so an approach whose range depends on a small board dimension is not appropriate. ## Constraints - `1 <= len(rectangles) <= 200_000` - `-10^9 <= x, y <= 10^9` - `0 < width, height <= 10^9` - Rectangles have disjoint interiors. - All input numbers are finite. ## Examples ```text Input: rectangles = [[0, 0, 4, 2]] Output: 1.0 ``` ```text Input: rectangles = [[0, 0, 2, 4], [10, 1, 2, 2]] Output: 2.0 Explanation: At y = 2, the first cake contributes area 4 below the line and the second contributes area 2 below it. The total area is 12, so both sides contain area 6. ```