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Very often, in order to realize a high-gain operation of gyro-traveling-wave tubes (gyro-TWTs), the tubes are designed as multistage amplifiers, in which each section is short enough and thus, stable. In such tubes, the wave amplified in the input and intermediate stages should be absorbed at their ends by matching loads, which often limits the average power capability of these devices. This limitation can be avoided if these stages operate in the regime of the wave absorption by an electron beam. A long time ago, such a regime in conventional TWTs driven by linear electron beams was studied by Kompfner (1950), so the effect of wave attenuation in the gain curve is known as the Kompfner dip. In the present paper, we consider a two-stage (gyro-)TWT in which the first section operates in the Kompfner dip regime. The small-signal theory is developed, which allows one to analyze the tradeoffs in the small-signal gain and bandwidth associated with this kind of operation. The large-signal simulations indicate that such a two-stage gyro-TWT can operate with the orbital efficiency of 40% (total efficiency of 15%) and the gain over 40 dB.