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Ascendant Challenges are weekly rotating time trials available within the Dreaming City. There are six different regions that are accessible via Ascendant Portals scattered throughout the City and its Lost Sectors, which are only made visible by utilizing a Tincture of Queensfoil.  Toland, the Shattered will often appear to lead the Guardian through the challenge. Petra Venj offers a weekly bounty for its completion.
The entrance to the Garrison is found in the final area of the Chamber of Starlight, on the ledge where a pair of Taken Hobgoblins lurk. This challenge involves sprinting through a twisting maze of bridges and floating stones, all while being hounded by Taken Thrall, Taken Hobgoblins, Shriekers, Taken Phalanxes, and Abyssal Champions. At the end of the winding ruins resides Drachna, Favored by Incaru, protected by more shriekers in a tight garden of large stones. Killing the Darkblade rewards Dreaming City gear.
Tucked in the ceiling of Aphelion's Rest just before the final encounter is the portal to Ouroborea, which deposits into a chamber filled with shielded crystals. The objective is to destroy the four Blights hidden in the outskirts, which will dispel the shields, but also spawn hordes of Taken Thrall. Destroying all the crystals without succumbing to the thrall or roving Abyssal Champions concludes the challenge.
This challenge is encountered on a cliffside within the Gardens of Esila, peeking out toward the central tree. Crow used to flick coins off this cliff to make wishes before he recognised the significance of the place.  Here the Guardian is required to collect four orbs of Taken power and bank them in a central pool, which is guarded by Taken Phalanxes, all the while avoiding the large Blights infesting the area. Three Abyssal Champions will stalk the Guardian throughout as well. Killing them and collecting all the Taken power will complete the challenge.
This challenge is well hidden in the Spine of Keres, all the way down by the shore. Thankfully, the influence of Queensfoil allows for a shortcut there. Jumping off the right side of the first bridge reveals a series of hidden platforms that lead down to the island just before the Ahamkara Lenses. Continue toward the rocky mists, and the portal is visible floating behind the Taken Vandal's vantage point. This challenge involves scaling a colossal statue of a Techeun, all the while being harassed by Taken Vandals and Taken Hobgoblins. At the top of the jumping puzzle awaits Val Eralus, the Obedient, who must be defeated to escape.
Keep of Honed Edges
Deep in the Harbinger's Seclude rests the grand statue of Osana Sov, overlooking the bay and infested with blight. Above her head rests the ascendant rift, which is accessible via a series of hidden platforms that spiral upwards, starting off the right side of the bridge approaching her. Entering the portal transports the Guardian outside a shattered ruin infested with Hive. Three former Princes kneel in the center of the chamber around a relic Cleaver. They will be invincible until the other Hive Wizards and Acolytes are cleared out. Taken reinforcements will spawn instead and the knights will stand, vulnerable. The sword can be used to slay the resurrected Swarm Princes. Then, the Guardian must escape back through the destabilizing debris field to where they entered to receive their reward.
Stashed in the Bay of Drowned Wishes, beneath the stone arch and immediately before the shielded cave is the rift into the Agonarch Abyss. A fitting name, as this challenge involves dropping down a dark chasm to the rotunda at the bottom, where Taken Wizards An Hamaat, the Warp and An Salaak, the Weft dwell. The latter will conjure Blights as area denial, which will disappear on her death. Killing both Wizards completes the challenge.