Oil, Water, Paths

Normal sudoku rules apply.

Each cage is either full of "oil" (even digits), full of "water" (odd digits), or a mix of both. In all cases, each "row" within a cage is of just one type (oil or water). Oil floats on water, so even digits will always be in rows above odd digits in mixed cages.

Digits in cages do not repeat.

Digits in cages must sum to the small clue, if given.

A path consisting of the digits 1 to 9, in order, crosses from left to right across the grid with each digit in its corresponding column (1 in column 1, 2 in column 2, etc.) The cells along the path are connected orthogonally or diagonally. A similar path crosses the grid from top to bottom, starting with a 1 in row 1, then a 2 in row 2, ending with a 9 in row 9. Again, cells along the path are connected orthogonally or diagonally.

Each path oscillates inside/outside cages, such that all of one path's oil cells (but not water cells) are in cages, and all the other path's water cells (but not oil cells) are in cages. It is permissible for a path to exit and re-enter the same cage.

The two paths (which do not share cells) must be determined by the solver.