Castle in the Sky

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In the film's backstory, human civilizations built flying cities, which were later destroyed by an unspecified catastrophe, forcing the survivors to live on the ground. The only remaining flying city, Laputa, still floats in the sky, concealed by a permanent powerful hurricane that surrounds it.
In the story's opening, an airship carrying Sheeta, a girl who has been abducted by Muska, a secret agent working for the government, is attacked by Captain Dola and her air pirate sons who are in search of Sheeta's crystal amulet. In the resulting struggle, Sheeta falls from the airship but her descent is slowed by the amulet which starts glowing. She safely lands in a small mining town where she is discovered by a boy named Pazu, who takes her into his home to recover. On waking, Sheeta is surprised to find that Pazu is building a small aeroplane with the intention of finding the lost city of Laputa, which his father had taken a photograph of while flying some years previously. She tells him that her family lived in a small valley to the north named Gondoa, and that she lived alone once her mother, father, and grandmother died, until Muska and his agents abducted her. Later that day they are pursued by Dola's pirates, and then by Muska's soldiers. Eventually, the two fall into an abandoned mine, where they encounter the local eccentric 'Uncle Pomme', who informs them that Sheeta's amulet is made of 'volucite' crystal ('Aetherium' in the American release), a material used to keep Laputa and the other flying cities aloft.[2]
Upon leaving the mines, Sheeta tells Pazu that her full name is 'Lusheeta Toel Ul Laputa'. They are then captured by Muska and taken to the fortress of Tedis, where Pazu is imprisoned in the fortress' dark, cold and damp tower while Sheeta is imprisoned in a more lavish room. Muska shows Sheeta a dormant Laputan robot and reveals his knowledge of her secret name, which he interprets to be that of the Laputan royal line. Muska then threatens Pazu's life to obtain Sheeta's cooperation. When they meet again, Sheeta orders Pazu to leave for his own safety, and Muska offers him money to leave and forget about Laputa.
A distraught Pazu returns home, where he is ambushed by Dola's sons. After being questioned, Dola explains to Pazu that Sheeta told him to leave in an effort to protect him. She also explains that once Muska has what he wants from her, he will most likely kill her. Upon hearing that the airship Goliath will be taking off to find Laputa with Sheeta aboard, Dola and her sons prepare to intercept and capture the crystal. Pazu asks Dola to join her pirates to save her. Dola initially refuses but reconsiders, and they immediately leave for the fortress to rescue Sheeta. As preparations proceed, Sheeta recites an apotropaic verse and unexpectedly activates the amulet and the robot, which follows Sheeta. When struck by Muska's artillery, the robot retaliates and proceeds to destroy the fortress. When Sheeta orders it to desist, the robot is overcome by Goliath. Pazu arrives and rescues Sheeta, but Muska obtains the amulet.
The pirates, accompanied by Pazu and Sheeta, return to their airship, Tiger Moth, where Dola assigns Sheeta to the galley and Pazu to assist the ship's engineer. They pursue the Goliath, which is following directions indicated by Sheeta's amulet to locate Laputa. Both airships arrive at Laputa on the following day. The two children, separated from Dola's pirates, discover the city to be devoid of human life, but overgrown by a park-like woodland with a gigantic "eternal tree of life" at its center, which is maintained by a robot similar to the one Sheeta encountered in the fortress.
Dola's pirates are captured and Muska's soldiers plunder the city's treasures. The ancient city is revealed to be twofold; a crumbling, overgrown, yet beautiful ruin of a castle above, and a perfectly preserved scientific marvel below. The city exemplifies the ultimate evolution of the crystal levitation technology displayed by Sheeta's amulet, as well as holography, magnetic cohesion, pseudo-nuclear weaponry and a veritable army of the semi-sentient robots. Upon gaining entrance to the city's central sphere, Muska captures Sheeta and his agents open fire upon Pazu, who escapes and frees Dola's pirates.
In the center of Laputa, which contains the immense 'volucite' crystal keeping the city aloft, Muska identifies himself as "Romuska Palo Ul Laputa", a member of another royal line, and he uses Sheeta's "key" crystal to access the advanced Laputian technology. He then massacres his own soldiers and destroys the Goliath. During the mayhem, Sheeta seizes her crystal amulet and flees, with Muska in pursuit. Encountering Pazu, Sheeta gives him her amulet through a gap in the wall and is cornered by Muska in Laputa's abandoned throne room. During her confrontation with Muska, Sheeta realizes and explains that the people of Laputa left the castle because they realized that man was meant to live on earth and not in the sky. Muska refuses her arguments, shoots off her braids and threatens to kill her unless the crystal amulet is given to him. Pazu then enters and says he'll give the crystal amulet to Muska if he can talk to Sheeta; Muska grants them one minute.
Sheeta and Pazu recite a "Spell of Destruction", destroying much of the city, which leads to Muska's death. Having survived the collapse, protected thanks to the tree's gigantic roots, Pazu and Sheeta re-unite with Dola and her pirates (who have had the presence of mind to swipe some of the city's treasures before fleeing) and leave Laputa behind. When they part with the pirates, Pazu flies Sheeta to Gondoa as he had promised her, to start a new life together. The end credits show the remnants of Laputa still floating in orbit, with the guardian robot still tending the garden, maintained by the volucite crystal embedded in the roots of the central tree.

Question: What lost city does Pazu wish to find with the aeroplane he is building?
Answer: Laputa