Jugs Riddle

Logic Level 3

I have 2 unmarked empty jugs of capacities \( \frac{2}{3} \) liters and \( \frac{5}{4}\) liters, as shown below left.

Because they are unmarked, I can only pour water from one jug to another until the initial jug is empty or the other is full. The unwanted water will be wasted into a sink, for there are no other containers.

After some pouring/filling/wasting, I measure exactly 1 liter of water in the bigger jug with the other empty, as shown.

Using the minimum amount of water to obtain the 1 liter, let \(x\) be the number of times an empty jug is filled from the faucet, and let \(y\) be the number of times a full jug is emptied into the sink. What is the value of \(x+y?\)

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