Top three of 25 horses with five-lane races

You have $25$ horses with distinct, consistent speeds and a track that fits $5$ horses per race. You only learn finishing order, not times. What is the fewest races that guarantees you know the $3$ fastest horses in order?

7. Five heats rank each group, and a race of the five winners finds the overall fastest. Only five horses can still be second or third: the 2nd and 3rd of the champion's heat, the winner and runner-up of the heat whose winner came second, and the winner of the heat whose winner came third. One more race among these settles it.