Inverted Gravity Grid Traversal

You are given an $R \times C$ grid containing empty spaces (.), solid walls (#), a starting position (S), and a target position (T).

You start at cell S with gravity pulling DOWN and $K$ gravity flips available. The outer perimeter beyond the grid boundaries (above row $0$, below row $R-1$, left of column $0$, and right of column $C-1$) behaves as solid walls.

Movement Rules & Physics

1. Gravity States: Gravity is either DOWN or UP.
2. Anchoring: You are considered anchored if:
   • Gravity is DOWN and the cell directly below you is a solid wall # (or row index $R$, the bottom boundary).
   • Gravity is UP and the cell directly above you is a solid wall # (or row index $-1$, the top boundary).
3. Actions:
   • Flip Gravity: At any time, if you have remaining flips ($k > 0$), you can toggle gravity between DOWN and UP. This uses $1$ flip and takes 0 time units. Anchoring status updates immediately.
   • Walk Left / Right: If you are anchored, you may walk $1$ cell left or $1$ cell right into an adjacent non-wall cell. This takes 1 time unit.
   • Fall / Float (Auto-move): If you are NOT anchored, you cannot walk left or right. You MUST move $1$ cell in the current gravity direction (DOWN or UP) into a non-wall cell. This takes 1 time unit.

Reaching the target cell T (whether by walking, falling, or floating onto it) immediately finishes the traversal. Determine the minimum total time units needed to reach T. If T cannot be reached, return -1.