RAW, can you break up turns between a rider and their controlled mount?

Can’t seem to find clarification on this, it always just directs me to the Mounted Combat rule.

Mounted Combat
[…] The initiative of a controlled mount changes to match yours when you mount it. It moves as you direct it, and it has only three action options: Dash, Disengage, and Dodge. A controlled mount can move and act even on the turn that you mount it. […]

Rules as written, can a rider attack, have the mount take the Disengage action and move, and make another attack on the same turn?