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Conceptually it's four latches, each controlled by one of the buttons, which inhibits the other latches when it is flipped. The outputs of the four latches can be or'd together to light a common light (but how do you indicate which button was pushed first?).
You'd need 4 lights and each light would have one JK-flip flop, and when the one JK-FF was set it would inhibit all other latches as well as activating the light that it belongs to (combine the active HI (or active LO) output of each JK-FF using an OR (or NOR) gate and run the output to the active HI (or active LO) clear signal of every other JK-FF for the inhibit function, and link all their reset signals together so you can restart). Inverters will be needed if the the active logic state of the clear pin is not the same as the active state of the JK-FF output.
You also have to make sure that the clear function of the JK-FF overrides the user signal to the latch for this to work properly (since they both could be activated at the same time) via the datasheet. THe clear and set signals on a JK-FF would probably have the same priority so you could not use the set signal for the user input. But it probably overrides the toggle signal so you could use that as the user input. As long as the JK-FF starts up with cleared output, toggling it after a reset is always equivelant to setting the output.
Extra drive circuitry controlled by the JK-FF output would be needed for the lights.