Chaos Construction: Region Counting

Rules:

Place the numbers from 1 to 9 exactly once in every row, column, and region. One number is already given. Each region is orthogonally connected and must be located by the solver.

A digit in a cell with an arrow indicates the total number of distinct regions between that cell and the edge of the grid in the direction of the arrow. All possible up, left, down, and right arrows are given.