Facility location is the problem of locating a public facility based on the preferences of multiple agents. In the classic framework, where each agent holds a single location on a line and can misreport it, strategyproof mechanisms for choosing the location of the facility are well-understood.We revisit this problem in a more general framework. We assume that each agent may hold several locations on the line with different degrees of importance to the agent. We study mechanisms which elicit the locations of the agents and different levels of information about their importance. Further, in addition to the classic manipulation of misreporting locations, we introduce and study a new manipulation, whereby agents may hide some of their locations. We argue for its novelty in facility location and applicability in practice. Our results provide a complete picture of the power of strategyproof mechanisms eliciting different levels of information and with respect to each type of manipulation. Surprisingly, we show that in some cases hiding locations can be a strictly more powerful manipulation than misreporting locations.