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Coz this button often do nothing. It is a "new" feature yet, implemented for future stuff, so girls have "actions" only in a few places.

What Aandred said. Basically, action is the button you hit in order to repeat a repeatable scene. For example, if a repeatable scene occurs for the first time in the morning in the living room, then it can be played again by talking to the same girl in the living room in the morning on following days. Talking to the same girl in the living room in the afternoon and hitting the action button will do nothing, because no event happened with that girl in that location in the afternoon.

Just reading it, it sounds convoluted and confusing, but in practice it's fairly straightforward. Experience an repeatable event with Lucy that you would like to see again? Speak with her in the same place at the same time of day and hit action, and the scene will repeat itself.