Water Level Through a Pyramid Apex

A closed container in the shape of a right square prism stands on a level surface. Embedded in its base is a solid decorative right square pyramid having the same base as the prism, with apex $P$ pointing upward. When the container holds $a$ litres of water, the water surface passes exactly through the apex $P$. When the container is turned upside down, the water surface again passes exactly through $P$. Consider the following four statements:

① The height of the pyramid equals half the height of the prism.

② When the container is laid on one of its side faces, the water surface still passes exactly through $P$.

③ In any orientation of the container, once the water settles the surface always passes exactly through $P$.

④ If a further $a$ litres of water is poured in, the container becomes exactly full.

Determine, with proof, which of these statements are true.