Nine unit circles are packed into a square, tangent to their neighbors and to the square. What is the length of the longest smooth path connecting two opposite corners of the square?

Assumptions:

- The path must be continuous and follow the lines in the diagram; that is, it must be made up of portions of either the circles or the outside square.

- The path may not change direction suddenly.

- The path may not contain any loops.

- The path may not touch or cross itself at any point.