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This paper considers a multimodal charging network in which charging stations and battery swapping stations are jointly built to support an electric ride-hailing fleet synergistically. Our argument is based on the observation that charging an EV is a time-consuming burden, and battery swapping faces scaling issues due to its deployment costs. However, charging stations are cost-effective, making them ideal for scaling up EV fleets, while battery swapping stations offer quick turnaround and can be deployed in tandem with charging stations to improve fleet utilization and reduce operational costs. To fulfill this vision, we consider a ride-hailing platform that jointly builds charging and battery swapping stations to support an EV fleet. An optimization model is proposed to capture the platform's planning and operational decisions. In particular, the model incorporates essential components such as elastic passenger demand, spatial charging equilibrium, charging and swapping congestion, etc. The overall problem is formulated as a nonconcave program. Instead of pursuing the globally optimal solution, we establish a tight upper bound through relaxation and decomposition, allowing us to evaluate the solution optimality even in the absence of concavity. Through case studies for Manhattan, New York City, we find that joint planning of charging and battery swapping stations outperforms deploying only one of them, yielding a total profit that is 11.7% higher than swapping-only deployment under a limited budget, and 17.5% higher than charging-only deployment under a sufficient budget. These results underscore the complementary benefit between charging and battery swapping facilities.