Check Vertical Reflection Symmetry with Duplicate Points

Given a multiset points of two-dimensional points, where each entry is an [x, y] pair with integer coordinates, implement has_vertical_symmetry(points: list[list[int]]) -> bool to decide whether the multiset can be mirrored across some vertical line.

Such a line has the form $x = c$, where $c$ may be an integer or a half-integer. Reflecting a point $(x, y)$ across the line produces $(2c - x, y)$. A point with $x = c$ maps to itself.

Duplicates must be preserved: for every point [x, y], its multiplicity must equal the multiplicity of its reflected point [2c - x, y]. The order of the input points does not matter. An empty input is symmetric by definition.

Example 1:

Input: points = [[2,5],[6,5],[4,1]]

Output: true

Explanation: With the line $x = 4$, (2,5) and (6,5) reflect into each other, while (4,1) lies on the line.

points = [[2, 5], [6, 5], [4, 1]]

true

Example 2:

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

Output: true

Explanation: The line $x = 0.5$ maps (-2,4) to (3,4) and (3,4) back to (-2,4).

Example 3:

Input: points = [[-5,1],[-5,1],[7,1]]

Output: false

Explanation: The line $x = 1$ maps (-5,1) to (7,1), but (-5,1) occurs twice while (7,1) occurs only once.

Constraints:

• $0 \le len(points) \le 150000$
• Every point contains exactly two integer coordinates.
• $-500000000 \le x, y \le 500000000$ for each coordinate.
• Repeated points are allowed, and their counts affect the result.