free and fixed pulley

hi guys,
Consider the well known system of one fixed and one free pulley.
A weight on the free pulley.
I know that my pulling force is half the force of gravity on the weight.
Why that is I understand.
What I do not get is why i have to haul in twice as much rope compared
to the change in height of the weight.

My suspicion is that it has something to do with the cycloid. Because
the free pulley rols upwards. I checked the formulas but I could not
find a factor 2.

thanks in advance, Peter
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