How many Roman Empires would it take to conquer the US? (They spawn randomly scattered)

The Roman armies do not appear as one massive invasion force, this is to avoid "Age of Empires" stream wars or nuke scenarios. Every legion, auxiliary force, cavalry unit, etc. spawns at random scattered locations throughout the geographical US. This means that they would spawn in the middle of the streets, inside offices, in the deserts, et-cetera, randomly scattered.

Win condition: the US government collapses/surrenders, the Constitution is abolished, and a Roman Emperor takes over as ruler of the United States. Victory requires actual occupation and governmental collapse, not just causing chaos.

Also:

• Romans have been briefed about this 1 week before, they've been shown pictures of the modern world and some basic facts about it to decrease their initial shock. When they spawn, they know about the win condition and they are magically motivated to reach it as their first priority.
• Romans do not require food or water, but still require sleep and normal human recovery.
• No immunity/disease issues.
• Peak Roman Empire under Trajan (117 AD)

There has to be some number of Roman Empires where the sheer amount of manpower becomes impossible to contain.