Check Vertical Reflection Symmetry with Duplicate Points

Determine whether a multiset of two-dimensional points is symmetric about some vertical line. Every occurrence of a point must have a reflected occurrence, including duplicate counts.

Function Signature

has_vertical_symmetry(points: list[list[int]]) -> bool

Rules

A vertical reflection line has equation x = c, where c may be an integer or a half-integer. Reflecting (x, y) produces (2*c - x, y).

Return true if there exists such a line for which each coordinate pair and its reflection have equal multiplicity in the input. Points on the line reflect to themselves. Input order is irrelevant.

For this exercise, the empty multiset is symmetric and returns true.

Constraints

• 0 <= len(points) <= 200000.
• Every point contains exactly two integer coordinates.
• Each coordinate is in [-1000000000, 1000000000].
• Repeated points are allowed and their counts matter.

Examples

Input: points = [[1,2],[3,2],[2,4]]

Output: true

The line is x = 2.

Input: points = [[0,1],[1,1]]

Output: true

The line is x = 0.5.

Input: points = [[0,0],[0,0],[2,0]]

Output: false

The left point occurs twice while its reflected point occurs only once.

Input: points = []

Output: true