How does breathing draw energy

Think of a cycle pump. When pulled, it draws air and when pushed it pushes air into the tube.

Analogous to this, the pump is our breathing, PULL is INHALE and PUSH is EXHALE but PUSH does 2 things here, Pump energy to the TUBE (BODY) and Pump exhaust air to the outside.

Very analogous to pumping a tube, while INHALing, the drawing in of air will / should be FAST and the PUMPing in / EXHALE will / should be SLOW