The Ice Cream Containers Problem

Here's a fun little math problem (for folks of any age) to play with.

There are three different containers of ice cream. Each container has 100 gallons of ice cream of either all Chocolate (in container X), all Vanilla (in container Y), or all Strawberry (in container Z). 

There are two robots with ice cream scoops controlled by an activation button. Each robot scoops a certain amount of ice cream from a container and deposits this into another container. Robot 1 simultaneously transfers 2/3 of the ice cream in container X to container Y, and 1/2 of the ice cream in container Y to container X. Robot 2 simultaneously transfers 1/6 of the ice cream in container Y to container Z, and transfers 1/3 of the ice cream in container Z to container X.

The activation button can be pushed again to repeat this process. Note that after the first button push, container X will no longer contain all Chocolate ice cream (it will actually contain 1/3 of the original amount of Chocolate ice cream plus 1/2 of the Vanilla ice cream originally in container Y plus 1/3 of the Strawberry ice cream originally in container Z).

Question 1: As the number of button pushes gets very large, how many gallons of ice cream are in containers X, Y and Z?

Question 2: As the number of button pushes gets very large, how much Chocolate, Vanilla, and Strawberry ice cream can be found in containers X, Y and Z?