waves - Energy conservation and interference

I have a problem with energy conservation in case of interfering waves.

Imagine two harmonic waves with amplitudes $A$. They both carry energy that is proportional to $A^2$, so the total energy is proportional to $2A^2$. When they interfere, the amplitude raises to $2A$, so energy is now proportional to $4A^2$ and bigger than before.

The equivalent question is what happens to the energy with the superposition of two waves that interfere destructively.

Also, if someone could comment on the statement about this problem in my physics book (Bykow, Butikow, Kondratiew):

The sources of the waves work with increased power during the interference because they feel the wave from the other source.