A. Closest Point

Consider a set of points on a line. The distance between two points 𝑖 and 𝑗 is |𝑖−𝑗|.

The point 𝑖 from the set is the closest to the point 𝑗 from the set, if there is no other point 𝑘 in the set such that the distance from 𝑗 to 𝑘 is strictly less than the distance from 𝑗 to 𝑖. In other words, all other points from the set have distance to 𝑗 greater or equal to |𝑖−𝑗|.

For example, consider a set of points {1,3,5,8}:

• for the point 1, the closest point is 3 (other points have distance greater than |1−3|=2);
• for the point 3, there are two closest points: 1 and 5;
• for the point 5, the closest point is 3 (but not 8, since its distance is greater than |3−5|);
• for the point 8, the closest point is 5.

You are given a set of points. You have to add an integer point into this set in such a way that it is different from every existing point in the set, and it becomes the closest point to every point in the set. Is it possible?