The hydraulic system handles move and operate (through a complex system of valves), all mechanisms mentioned above. (Also used in brakes and other parts of the car)
The hydraulic pump is in the engine.
The advantage of this system is that it occupies little and is very effective and precise.
The biggest disadvantage is that due to the temperatures that occur inside the car and the constant and strong vibrations that must withstand the failure of the hydraulic system is very common in F1 cars