General aviation pilot reported a NMAC in the traffic pattern resulting from failure to follow traffic as instructed by ATC.

Narrative

The tower was very busy and was using the south frequency to work both runways. I was doing touch and go's and was on a left downwind for Runway XXL. The tower called; Cirrus; follow Cessna ahead and to the right number 3 Runway XX left cleared touch and go." I did not hear the full transmission so requested; 'Say again please.' The tower responded; 'Cirrus follow Cessna ahead in the base turn Runway XX cleared touch and go number 3.' I responded; "XXL touch and go number 3." I did not call traffic in sight. I was reducing power; lower the flaps and looking for the traffic that had been called out in their base turn when I saw the Cessna to the right; slightly in front of me and crossing right to the left and several hundred feet below me. This was almost immediately after I had read back the instructions. They were on their base leg. There was no time to maneuver to cross behind them. I extended my downwind leg in order to put some distance between us before I turned my base leg. I believe conflict was caused by the combination of my missing most of the tower's first call while I was on the left downwind and then when the tower repeated the call; the ability to widen my downwind from the airport to allow me to fall in behind the Cessna was lost as we were too close. I did not have the traffic in sight until it was crossing below me from right to left."