The rank of "count" was originally comte and this derives from Latin comitem "a companion". We can analyse this as the suffix com- "together" and itum, the supine of the verb √i "to go". The supine is a form of action noun in Latin. And a county was originally a province governed by a count. The word country is however unrelated and comes from Vulgar Latin (terra) contrata: The land lying opposite (contra).
Counting as a verb looks related by is actually from an entirely different root. To Count comes from Old French conter "add up" which is itself from the Latin computare "sum up, reckon". The root here is putare originally meaning "to prune".
While country looks related and isn't, circuit doesn't look related but is. Circuit comes from Latin circummitus, i.e. circum "around" and ite "to go" (also from √i).
Where do police come into it? Well, a junior police in Britain is a constable. This comes from Old French conestable (which in modern French is connétable). This again comes from Latin comes stabuli. Here, comes is the same as comte (and thus ultimately from √i) and means "a count" (the rank), while stabuli means "stable". Count of the stable was the title of a dignitary in the Roman Empire and was established by Codex Theodosianus (ca. 438 CE). In essence this is a fancy way of referring to the head groom. However, the title was kept in use by the Frankish kings and referred to stewart of their household. In English it referred a junior legal officers, and then in 1836 was first used to refer to a police officer.
So counts, circuits, and police all have the verb √i "to go" in common. And the connection to Monty Python is via the grammar lesson scene in The Life of Brian. Brian is commissioned to graffiti "Romans go home" on the Roman headquarters. He paints Romanes eunt domus "Romans they go the house". But he is caught by a patrol and the centurion critiques his Latin grammar and forces him to conjugate the verb √i at the point of a sword: the grammatically correct sentence is Romani ite domum. I checked and this is one phrase that Google translator gets right!
