WI: Heraclea Pontica wank/Achaemenids restored?

Heraclea Pontica was an ancient Greek city on the coast of Bythnia, near the river Lycus, founded by Megara in 560-558 BC, near the Black Sea, and was named after the place where it was believed Heracles entered into the underworld through a cave. The colonists quickly subjugated the native inhabitants, known as the Mariandynians, and extended their control over the area. It later became a tyranny when Clearchus, a citizen of the city, after making a deal with Mithridates of Cius and reneging on it later, obtained command of a body of mercenaries and made himself tyrant, in 365 BC. While he was eventually assassinated, one of his sons, Dionysus, after the reigns of his uncle and brother, assumed the tyranny in 337/336 BC, and died in 306/305 BC. Now we go and focus on what was happening on a much larger scale. After Alexander defeated the Achaemenids and conquered Persia, many members of the former ruling family, the Achaemenids, found themselves spared and treated well by the conqueror, including the brother of Darius III, Oxyathres, who had a daughter known as Amastris. Amastris was initially given in marriage to Craterus at the Susa weddings, but Craterus divorced her in 322 BC, to marry Phila, the daughter of Antipater, before his own death at the Battle of the Hellespont. Amastris then married Dionysus, and had two children, Clearchus II and Oxyathres, who eventually killed her in 284 BC. Dionysus died in 306/305 BC, and Amastris became guardian of the children, ruling in her own right, before marrying Lysimachus in 302 BC, though Lysimachus would abandon her shortly afterwards to marry Arsinoe II, one of the daughters of Ptolemy. Once her marriage to Lysimachus ended, she ruled Heraclea Pontica in her own right. What we have, here, is an Achaemenid princess ruling a heavily Greek city, a direct descendant, many years after the Achaemenid empire itself was defeated and destroyed, unlike the dubious claims made by the monarchs of the Pontic kingdom. So how can we wank the kingdom/tyranny ruled by Amastris, a direct descendant of the Achaemenids, the noble and illustrious Cyrus and Darius the Great? What if Lysimachus doesn't decide to marry Arsinoe, or Ptolemy doesn't offer, and Lysimachus dies at Ipsus, and she successfully manages to position herself as ruler of Thrace? Can this kingdom take parts of Anatolia and Hellas, or go for easy expansion in the north like what Pontus did? If we go with an extreme wank, could we see the Achaemenid empire restored, albeit with heavy Hellenistic influence? Rome managed to grow from a small settlement to an empire ruling one-fifth of the world's population, spanning from Britain to Mesopotamia, and improbable, implausible, or even outright ASB things(from the perspective of other TL's) have happened in history before. What would be the impact? Could it defeat the Seleucids and deal with the Celtic migrations? Could it even retake Persia? How would such an empire function, and what would it look like?
 
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There's even more incentive in this case to Hellenize. Undoubtedly the elite were already in a small sense "Achaemenid Greeks."
So it would function largely as a Hellenistic kingdom with only the royal bloodline itself being different, instead of going back to their Achaemenid roots in a more concrete manner than a few propaganda displays on coins and statues?
 
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