Reversing Clutch of Two Velocities

Input: pink shaft of a gear and a pulley rotating one way (clockwise) at Vi velocity.
Output: grey shaft that can rotate two ways subject to axial position of the orange friction clutch.
When the clutch is in the left, the output rotates counterclockwise at velocity V1
V1 = Vi/((Z2Z4)/(Z1Z3))
Z1, Z2, Z3 and Z4 are tooth numbers of pink, large green, small green and violet gears respectively.
In this video Z1 = 20; Z2 = 80; Z3 = 20; Z4 = 60 so V1 = Vi/12
When the clutch is in the right, the output rotates clockwise at velocity V2 = Vi (together with the input).
The grey shaft has prismatic joint with the orange clutch.

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