Shut the Box

You are given a 6×6 grid. Some cells carry clues:

Rules.

1. Cut away one or more groups of orthogonally connected cells. Every group you cut away must include at least one cell on the border of the grid. The cells that remain (the region) must be orthogonally connected and have no holes.
2. It must be possible to fold the region along grid lines so that it forms the six walls of a rectangular box. Each cell of the region becomes one unit square of the box's surface, and every unit square of the surface is covered by exactly one cell (no overlaps, no gaps). A face of the box need not come from one connected piece of the flat grid: its cells may be split across different parts of the region.
3. An arrow cell is not in the region. Look along its row and column. Among the four directions, the arrowheads show exactly the direction(s) in which the nearest region cell lies (if two or more directions tie for nearest, all of them get an arrowhead). All three arrows here have only a left arrowhead.
4. A number cell is in the region. Every number is a number cell, including the numbers in grey circles. The number counts the region cells in the 3×3 block centred on it (the cell itself included; cells outside the grid do not count).
5. When the box is assembled, each grey circle must be directly opposite the other grey circle: the two circled cells lie on opposite (parallel) faces, and the line segment joining their centres is perpendicular to both faces.

There is exactly one valid region, and it folds into exactly one box.

Task. Assemble the box. On each of its six faces, add up all the numbers on that face (plain and circled alike). Your answer is the product of the six face sums.

Answer format: a single integer (for example 12345).