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Cicada 3301 mentions within the first post they made "Your pilgrimage has begun. Enlightenment awaits." Pilgrimages are normally a religious journey, most commonly there is Haji. Enlightenment is a philosophical and religious experience. They have an obsession with the numbers 3301 and 1033 and consistently reference various older religions often related to treating oneself as their own god. Recently while researching Freemasonry someone mentioned Shriners would often wear a fez representing intelligence among other things and is traditionally red and made of felt. When I looked up 1033 on Wikipedia I found "Fez Massacre" which occurred in 1033. The Jews were massacred in this. On 1033 Anselm of Canterbury was born and is well-known specifically for his philosophical/ontological arguments proving God's existence as best he could through reasoning of the time, saying basically that God would be a various set of the best thinkable traits and be the highest quality possible. They've notably shown they're looking for highly intelligent individuals and applied heavy cryptography and Fezes are known for symbolizing intelligence Putting this information together there is a string of related information: Religions, Shriners, Fezes, Fez Massacre, 1033, Jews, Haji, God, Intelligence, Enlightenment
From this string of information if it is all related, it can be deduced the organization would likely be religious, elitist (intellectually), clearly secretive due to the cryptography, like Judaism, like Fezes, and are obsessed with the numbers 3301 and 1033.
What organizations exist that are highly elitist when it comes to knowledge, enjoy the teachings of older religions, are secretive, have an especially favorable view of Judaism, would have a favorable view of Haji, are related to Shriners, desire enlightenment and like Fezes?
The Shriners. This may be a new secret branch of masonry or the Shriners, but the connections are there.
And if not those two things, I will now reasonably posit based on the above information this is in fact a new-age version of the Illuminati. I'm not trying to make a joke in positing the notion, it's just when one considers how the illuminati would realistically exist, this would make sense as a form they could take. It'd be an argument of semantics if they really do call the thing Cicada 3301 but if you remove the title and look at the content, the most fitting known title would then be The Illuminati.
TL;DR Illuminati is officially confirmed.