You manage a barrel warehouse. The warehouse is divided into an $8\times 8$ grid of cells. A barrel takes up one cell when upright, and two cells when laid down. You only remember where some of the barrels are in the warehouse, and you must determine where the rest of them are. Luckily, you remember two things that should help you out:
- There are indicators along each row/column representing the number of barrels that occupy at least one cell in that row/column.
- Barrels that are lying down have no room to roll to either side, i.e. there is a barrel or the wall stopping them from rolling on either side.
Here is a diagram of the warehouse, with the row/column indicators and the barrels you know for sure. Please determine where the remaining barrels are, and show your steps.