How are both squawk and altitude transmited when operating a Transponder in Mode C?

I know that in Mode $$3$$ (which includes Mode A and Mode C) the aircraft transmits $$12$$ information pulses. That corresponds to $$2^{12}=4096$$ squawk combinations in Mode A which are sent to ATC to identify the aircraft. In Mode C, all $$4096$$ combinations are used to send the pressure altitude of an aircraft to the ATC.

Since there are $$12$$ usable information pulses, when operating in Mode C, there is no way to transmit the squawk code, which in my mind raises an obvious question: What if I suddenly squawk $$7700$$ due to an in-flight emergency? Or $$7600$$, radio failure? If I continue to operate in Mode C, it seems that there is no way that ATC would know that I am in any kind of trouble. How is this issue resolved and is my understanding correct?

• Mode 3/A and Mode C are not the same thing. – StephenS Sep 20 '20 at 15:26