Can the instantaneous power output of an ac source ever be negative? Can the average power output be negative?

Suppose we have a capacitor attached to an ac source (there is no resistor). When the capacitor is charging, the power output is increasing and is positive. Then, capacitor starts discharging but the power output is still positive yet decreasing. Then, as the polarity of the current changes and the capacitor starts charging in the opposite direction. Power is increasing and in negative. Hence, instantaneous power can be negative.

Now, consider average power.

where Z is the impedance and ϕ is the phase angle. Impedance is always non-negative and square of rms voltage is also non-negative. Also,

where R is the resistance. As R and Z are always non-negative, cos(ϕ) is always non-negative. Hence, average power cannot be negative.