The actions that would be most likely to affect the accuracy of the compass are:
A. Moving the compass close to a magnet - This would directly interfere with the compass needle's alignment with the Earth's magnetic field.
B. Holding the compass next to rocks containing iron - Iron can create a local magnetic field that can disrupt the compass's readings.
E. Walking under overhead power lines - Power lines can generate electromagnetic fields that may affect the compass accuracy.
So, the correct choices are A, B, and E.