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NEN Hunter x Hunter abilities operate on a ridiculously complex system called 'Nen' - it's a technique that allows a living being to use and manipulate their own life energy (known as aura). There are principles and a person's own individual use of it, I'll try my best to be brief and focus on in-game uses rather than the technicalities, but I can explain further in case it's needed! Chrollo is also absurdly OP, so I am also giving some nerfs to his abilities!
GENERAL USES OF NEN

TEN is like an aura shield, it allows the user to control their aura to keep it from leaking, preserving energy, and as a protective layer against aura attacks and to some degree, physical attacks as well.
ZETSU is the aura concealer, it allows for one's presence to be undetectable, but leaves the user vulnerable to attacks. Think of that feeling of someone watching you - Zetsu kills it.
REN is the aura amplifier, it projects it out and enhances their physical and Nen abilities. If the user is filled with hostility, it can be felt as a palpable and paralyzing 'bloodlust'.
GYO allows for the user to concentrate their aura on their eyes, and as a result, it allows them to detect things otherwise invisible to the naked eye.
IN is a ZETSU but stronger, it allows for one's presence to be completely invisible, but leaves the user vulnerable to attacks.
EN is an aura expander, as the aura extends around a spherical shield and allows the user to sense what's in that area - it's for monitoring, basically.
SHU is a technique that enables the user to extend their aura into an object, which imbues it with a person's will and as an extension of oneself.
KO concentrates all of a person's aura into a specific part of the body, leaving the rest completely vulnerable, but also enhances that body part dramatically.
KEN is the opposite - it's basically TEN, but more uniform and stronger.
RYU is a technique that involves the continuous shifting and reallocation of aura, seamlessly allowing for all the above to be used naturally.
HATSU is an unique use of one's aura, their signature, which is deeply personal and reflects a user's personality, aspirations and even fears, as one's most personal self-expression there is. They all have different types of aura, which I'll leave here, there's math involved, it's a whole thing! In Chrollo's case, he's a Specialist, a rare type of Nen, and his ability is named Bandit's Secret.

BANDIT'S SECRET

With his ability, Chrollo conjures a book named "Bandit's Secret", and by fulfilling strict conditions, he's can steal one's hatsu, leaving the victim without access to their ability. The conditions are that Chrollo must witness the Nen ability in action with his eyes, ask questions about it and be answered, and the victim's palm must touch the handprint on the book's cover. To use an ability, Chrollo has to materialize the book, turn to the page of the desired ability, and it must remain open in his right hand, which allows him to use one ability at a time without using his bookmark. The abilities are subjected not only to the limitations but also to the emotions and thinking of their creators, so he might have to bend them to his use. Should the original user die, with some exceptions, the ability disappears from the book.

When abilities require both hands, Chrollo can conjure a bookmark to allow him to maintain access to any ability on the page it is placed, even if the book is closed and he is not touching it, acting as a loophole. Furthermore, if he keeps the book open on an ability page and he holds it and uses the bookmark on another, he can use two stolen abilities at the same time.

Closing the book without the use of the bookmark ends the use of the ability.

As Nen is exclusive to Hunter X Hunter, I assume he won't be able to steal new abilities in Etraya, unless they're castmates, and only should they permit me to do so.

STOLEN ABILITIES AVAILABLE

Indoor Fish are huge skeleton fishes that can only exist in completely sealed rooms, and feed on human flesh. While a person is being eaten, they don't feel pain or bleed and will remain alive and completely conscious during it. When the fish disappears, the effects of the wounds manifest, which likely kills the poor target.

Fun Fun Cloth is a cloth that can increase and decrease in size, and whatever is covered with it will shrink until it fits the palm of the user's hand, and can be returned to the original size by being let out of the bag. People can also be shrunk, and not be injured. We don't have the exact limitation of how many objects it can hold, nor about size - but I will always make sure to double-check the possibility of it should I want to use it on something larger than a car!

Sun and Moon create two symbols on the back of Chrollo's hands, his left is a sun with a plus sign, and the right is a moon with a minus sign. by touching a person, he can mark that individual with the symbol on the hand he used, and when two marks touch, they trigger an explosion. While they can be created instantly, to reach full potential, Chrollo needs to touch them for 3-5 seconds. A fully charged explosion is enough to kill people around the person marked, and the fully charged symbols were able to turn what they were affixed on into a bomb, including parts separated after the application of the symbols. Since it is one of the exceptions I mentioned before and the owner was very mad, the ability remained in the book after he died, and it can bypass restrictions of other abilities, as it doesn't disappear from a person's body when Chrollo closes the book.

Order Stamp lets him conjure a stamp that allows the user to control a puppet that they stamp by giving a vocal command. The thing here is that to be considered a puppet, the object the user wishes to control must have a head and be lifeless - but not corpses, as the original user didn't view them as inanimate objects, but it can control copies of corpses! The stamp has the kanji for "person" (人) and leaves the same mark upon being affixed, and while it is capable of controlling a ridiculous (+200) number of puppets, the orders can't be complex. If controlling copies of corpses, he also has to keep in mind the morals and ideals of that person, as he can't just say 'kill that person' to a copy of someone who wasn't keen on it, as it might go ignored, so instead, he'd have to say 'break that person', for instance.
The nerf I suggest for this is a limit of five puppets at a time.

Convert Hands brings two marks on the palms of Chrollo's hand. Touching a target with his right-hand causes the target to take Chrollo's likeness, and touching them with the left causes them to take Chrollo's likeness instead. If he touches a target with both hands, he instantly switches appearances with that person. The marks stay on Chrollo's hands so that he can be identified. It is one of the abilities that require the bookmark, but after activating it, Chrollo can hold the book as normal.

Black Voice is an ability that allows the user to control another person from a phone by sticking an antenna on them. The user gains complete control over mind and actions of a person by texting commands on the phone until the antenna is removed or the person dies. There are two antennas, which can control two people at a time, and look at that, he can even put them on auto-pilot. By sticking an antenna on oneself, the user can actually put themselves in auto-pilot and gain incredible strength and speed until they complete the command, except it causes intense muscle pain for days and leaves the user not remembering what he had done while in it. As of his canon point, Chrollo has the phone, but has lost the antennas.

He comes from Meteor City, a city full of residents who aren't documented, recognized, or even known about, ghosts living amongst trash that other nations leave there, and a site of horrific deeds by outsiders against people they don't consider humans. As a joyful, intelligent, and curious child, Chrollo wanted to be an actor and perform with his troupe of friends, something he started doing to bring some hope, fun, and a sense of community to the other residents.

That changed when outsiders got to one of them, and he decided then to live his life as a villain alongside his friends, so violently rotten that no one would take anything from his hometown again. From then on, he embodied that act so much so that he no longer understands who he is underneath it, as fate is inevitable that way. This is something he truly believes in, destiny, souls, and fate, to the point he does not consider Judas a traitor, as if everything is pre-determined, Judas was simply playing a necessary role in Jesus' story. He reiterates it, when he mentions that he finds humans to be equals as puppets, and yet finds some admiration for Hisoka for trying to defy the inevitability of his death when he fights Chrollo. Again on the theme, he steals a prediction ability with a 100% accuracy rate, as he yearns for insight on where to go, what to do, what exactly is the best for him, and how to keep his people from finding misfortune.

The Spider, or the Phantom Troupe, is a large chunk of his identity due to the philosophy he created. He is a part of the Spider, but the Spider is all of him, and when he loses a part of it, he is distraught as if he lost part of himself, so he cares little about his life, dances with death, and accepts his own because it is hardly important when the Spider lives on. As the leader of the Spider, while Chrollo has always been extremely intelligent, interested in books and languages, and resourceful in learning, he is narrow-minded when focused on a goal, and almost naive when it comes to emotions. He expected his people to adhere to the rule that he is disposable and put his life in danger, forgetting that even if the Spider is absolute, he remains a loved person to them.

Perhaps because he has been robbed of several precious things, his role as a thief is so important to him. He cherishes his stolen items and admires them before he sells them, and there is no interest in money motivation for in it for him. It's the act itself, almost like giving back what he was given, in all he does. Even his ability is a reflection of this, as it allows him to steal a person's deepest self-expression and make it his own in use, restricting a person of their signature.

Finally, he's extremely malleable. His voice is quiet and low, his presence if not for the attire and overall evil vibe can be bland, and he's direct and eloquent when speaking - and yet, he will be whoever he needs to be in that moment, and he will give the performance of his life through it.