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We develop a methodology for the design of multiple-cavity klystron interaction circuits. We demonstrate our approach with the detailed design of a collector and a four-cavity circuit for a multiple-beam klystron (MBK) operating in the fundamental mode at a center frequency of 3.27 GHz (S-band). These elements are designed to be used with a 32-A 45-kV magnetically shielded eight-beam electron gun currently under fabrication . Upon integration of the gun, circuit, and collector, the MBK will be used for beam transport and beam-wave interaction studies and to validate developmental design codes and design methodologies. The device has a predicted gain of 33 dB at a peak pulsed output power of 750 kW with a corresponding electronic efficiency of 52%. For the present design, broad bandwidth is not a design objective, and the 3-dB bandwidth is 2.5%. Downstream of the output cavity, the magnetic field profile and the interior surface profile of the collector are carefully shaped to minimize the space-charge potential depression at the entrance to the collector, minimizing reflected electrons. The maximum calculated instantaneous power density on the walls of the collector is approximately 55 kW/cm2; at low duty cycles (<1.8%), the average power density is well within the limits for liquid cooling for pulse lengths up to 1.3 ms.