Roman vs. Mongolian Empire (+time-travelling vets)

Which of these ancient superpowers would win a confrontation?

Assume that...

-They are as advanced in technology and in combat technique as they were in their most powerful times (~200 AD and ~1250 AD, respectively)

-They are both highly motivated and bloodlusted

-They get several days of preparation time before the confrontation

-If they are able to loot some technology used by the enemy but unknown to them, we should assume that they learn how to use it, but only with 50% efficiency.

-On both sides, about 15.600 men are fighting, equaling the size of three Roman legions each.

Of course, the biggest advance of the Romans may be their incredible weapon technology with catapults and "scorpions" as well as their discipline and their very sorted fighting formation like the "testudo formation".Additionally, they were very advanced in medicine for their time.

The Mongolians, on the other hand, are also not too bad-equipped and may even have some first, however really primitive fireweapon-prototypes.Their biggest strength, however, is probably their strong cavalry.

Round 1 (Roman advantage) : The fight takes place just outside their beloved capital - resulting in a moral boost as well as a huge strategical advantage for the Romans since they are able to get in defilade and can also fight downhill because of the topographical features of Rome.

Round 2 (Completely balanced) : The fight takes place in a neutral surrounding, i.e. in some environment both combattants are either familiar with or both combattants are equally unfamiliar with.There are no extraordinary topographical features.

Round 3 (Mongolian advantage) : The fight takes place in the Central-Asian plain, the topography is absolutely perfect for fighting on horses.The Mongolians do also have two small, but potentially effective gimmicks :

1.Their bows do get a modern sight in order to be able to aim more precisely

2.A group of a dozen modern-day vets joins them to care for the horses' health

The battle ends as soon as all fighters from one side have died.

It may, however, be worth a thought that the Mongolians gave up on conquering Austria in 1241, when their leader, Ögedei Khan, died. If the Romans knew this, they would therefore most likely aim on the Khan himself - but same may be true for the Roman emperor.