Why did the incans build such a large road network in their empire?

The Incas built such a  large road network stretching for 1000's of kilometers north and south of Cuzco, their capital, in what is now Peru, in order to keep the lines of communication open with their far flung empire, ie to convey messages to their local rulers, and probably for military purposes to have routes for their armies to follow in times of war. The roads were built by the ordinary people ie the men of  a certain age would be subject to the mit'a or compulsory public service to build roads, temples and work in the mines.

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