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I must clarify that there is no historical record of the Göktürks laying siege to Chersonesus (an ancient Greek colony located in modern-day Crimea) in 581 CE. The Göktürks, a nomadic confederation of Turkic peoples, were primarily active in Central Asia and the steppes north of China during this period. Their focus was on consolidating power in the region and engaging in conflicts with the Eastern Roman (Byzantine) Empire, the Sassanian Empire, and other neighboring powers.
Chersonesus, on the other hand, was a significant Byzantine outpost in Crimea and a key trading hub. While the Göktürks did have interactions with the Byzantine Empire, including diplomatic exchanges and occasional military alliances, there is no evidence to suggest they directly besieged Chersonesus in 581 CE.
If you are interested in the Göktürks or the history of Chersonesus during this period, I can provide more context on their respective histories and interactions. Let me know how you'd like to proceed!