Nurimisaki

A rectangular or square grid contains circles in some cells; a circle may have a number inside or not. The task is to blacken some cells of a grid according to the following rules: White cells form a path exactly one cell wide.Cells with circles are always white.A cell with a circle must have exactly one white cell adjacent to its side.A white cell without a circle must have at least two orthogonally adjacent white cells.A number in a circle indicates how many white cells can be seen horizontally and vertically from that cell, including the cell itself.All the white cells must be connected horizontally or vertically.No 2 x 2 cell area within the grid can have the same color