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Once the nation is a vassal, check to see if the vassal has any core provinces held by other powers, and plan on get them back to the vassal before annexing them. A nation stays a vassal even if it grows bigger than the initial ceiling for being a vassal. Also, selling provinces to a vassal if they have a claim or a core on it and if its not yet cored it is a good tactic. After a minimum 10-year wait, the player is free to annex any vassal with a relationship of 190+ or more. Even if a vassal changes religion or government type and would be impossible to vassalize through diplomacy, the player can still annex any vassal using a diplomat.
Considering the role of diplomacy in pursuing this achievement, the diplomacy idea tree is a very good choice. It lets the nation have more diplomatic relationships which means more simultaneous vassals. Also, more diplomats to improve relations, fabricate claims, and actual annexation.
The 奥斯曼 are a strong choice to pursue this achievement with. It has a national idea that reduces coring costs by 20%. Combine this bonus with the administrative idea adaptability (25% reduction) and having a diplomat create a claim on a province (10% reduction), the player will only pay 32% of the base cost in administrative points to core a conquered province. This means military expansion is a lot cheaper. Furthermore, the Ottomans have a lot of lower technology targets to the east and south-east, which means easier expansion.
Another option is 俄罗斯 (via 莫斯科), due to the large number of Siberian provinces, easily conquered hordes, possible colonisation of the American West Coast and potential conquest of China. A fun one can be 法兰西 by first doing achievement Big Blue Blob and then continue and take 100 provinces more.
For other game styles this can also be achieved by aggressively colonizing in the North Americas as the 阿兹特克s and relieving the 印加 and North American tribes of their lands. The player would do well to wall up the coastal provinces and than work their way inwards, take the Galapagos and Easter island to help prevent further colonization by the Europeans. Annexing the Inca will also give the player the achievement A Pile of Gold for owning 10 provinces of gold.
Finally, a player can play as a released colonial nation, such as the 美利坚, and colonize the interior of the Americas. This can result in getting the Aggressive Expander achievement without aggressively expanding at all.