One Flew Over the Cuckoo's Nest

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In 1963 Oregon, Randle Patrick "Mac" McMurphy (Jack Nicholson), a recidivist anti-authoritarian criminal serving a short sentence on a prison farm for the statutory rape of a 15-year-old girl, is transferred to a mental institution for evaluation. Although he does not show any overt signs of mental illness, he hopes to avoid hard labor and serve the rest of his sentence in a more relaxed hospital environment.
McMurphy's ward is run by steely, unyielding Nurse Mildred Ratched (Louise Fletcher), a strict authoritarian woman who passive-aggressively employs subtle humiliation, unpleasant medical treatments and a mind-numbing daily routine to suppress the patients and keep them submissive. McMurphy finds that they are actually more fearful of Nurse Ratched and her methods than they are focused on their main goal of eventually becoming functional in the outside world. In order to help them achieve this (as he believes), McMurphy establishes himself immediately as the leader of his fellow patients. The ward's other patients include Billy Bibbit (Brad Dourif), a nervous, anxiety-prone young man with a noticeable stutter; Charlie Cheswick (Sydney Lassick), a man disposed to childish fits of temper; Martini (Danny DeVito), who is delusional; Dale Harding (William Redfield), a high-strung, well-educated paranoid; Max Taber (Christopher Lloyd), who is chronically belligerent and profane; Jim Sefelt (William Duell), an epileptic man; and "Chief" Bromden (Will Sampson), a silent Native American-descended man of very imposing stature who is believed to be both deaf and mute.
Nurse Ratched soon comes to see McMurphy's newfound presence to be a threat to her authority and total control of the ward's patients, McMurphy's and Ratched's battle of wills escalates rapidly. When McMurphy begins hosting card games and wins away the patients' cigarettes, Nurse Ratched confiscates them and then begins to ration them out. Further challenging her authority, McMurphy calls for votes on such matters as policy changes or being able to watch the World Series. He also boasts to his fellow patients by betting that he can escape the facility simply by lifting an old hydrotherapy console (a massive marble plumbing fixture) from the floor and heaving it through the window, but fails to do so.
McMurphy then steals a hospital bus and herds his colleagues aboard, stops to pick up a party girl named Candy (Marya Small), then takes the group deep-sea fishing on a commandeered boat. He tells them (in a famous line from the film), "You're not nuts; you're fishermen!" As a result, the patients begin to feel faint stirrings of self-determination.
Soon after, however, McMurphy learns that Ratched and the doctors have the power to keep him committed indefinitely. Sensing a rising tide of insurrection among the group, Ratched tightens her grip on everyone. During one of her group therapy sessions McMurphy learns most of the patients, with the exception of Bromden, Taber, some of the chronics and himself, are voluntarily committed and are permitted to leave anytime they choose. Cheswick's agitation boils over about his confiscated cigarettes, and he, McMurphy and the Chief wind up brawling with the orderlies. As punishment for the incident, the three are all sent to ostensibly undergo treatment in the "Shock Shop" (a euphemism for electroconvulsive therapy, or ECT). While McMurphy and the Chief wait their turn, McMurphy offers Chief a piece of gum, and Chief murmurs "Thank you...Ah, Juicy Fruit." McMurphy is delighted to find that Bromden is neither deaf nor mute; he remains silent simply to deflect attention. After being subjected to ECT, McMurphy shuffles back onto the ward feigning brain damage, but then begins humorously animating his face and loudly greeting his fellow patients, assuring everyone that the ECT only charged him up all the more.
As the struggle with Ratched takes its toll, and with his release date no longer a certainty, McMurphy plans an escape. He phones Candy to bring her friend Rose (Louisa Moritz) and some booze to the hospital late one night. They enter through a window after McMurphy bribes the night orderly, Mr. Turkle (Scatman Crothers). McMurphy and Candy invite the patients into the day room for a Christmas party; the group breaks into the drug locker, puts on music, and celebrates. At the end of the night, McMurphy and Bromden prepare to climb out the window with the girls. McMurphy says goodbye to everyone, and invites an emotional Billy to escape with them; he declines, saying he is not yet ready to leave the hospitalâthough he would like to date Candy in the future. McMurphy insists Billy have sex with Candy right then and there. Billy and Candy agree and they move to a private room. Eventually the combined effects of the alcohol and the pilfered medication take their toll on everyone, including McMurphy and the Chief, both of whose eyes slowly close in fatigue.
Ratched arrives the following morning and discovers evidence of the party the night before; the ward is completely upended and the patients are passed out all over the floor. She orders the attendants to lock the window, clean up, and conduct a head count. When they discover Billy and Candy (having spent the night together after having sex), the other patients applaud Billy for doing so. Buoyed by the admiration of his fellow patients, he is able to speak without a stutter for the first time. However, in order to regain control of Billy and erode his newfound confidence, Nurse Ratched announces that she will inform Billy's mother what he had done. Billy's confidence immediately vanishes and his stutter returns as a result of her threat. Soon Billy starts punching himself and subsequently locks himself in the doctor's office. While alone in the office, Billy kills himself. As a result of this, McMurphy, distraught and enraged at Nurse Ratched for driving Billy to suicide, attacks her and nearly strangles her to death before an orderly, Washington, knocks him out.
Some time later the patients in the ward are seen playing card games and gambling for cigarettes as before, only now with Harding dealing and delivering a pale imitation of McMurphy's patter. Ratched, still recovering from the neck injury sustained during McMurphy's attack, wears a neck brace and speaks in a thin, reedy voice. The patients pass a whispered rumor that McMurphy dramatically escaped the hospital rather than being taken "upstairs".
Late that night, Bromden sees McMurphy being escorted back to his bed, and initially believes that he has returned so they can escape together, which he is now ready to do since McMurphy has made him feel "as big as a mountain". However, when he looks closely at McMurphy's unresponsive face, he is horrified to see lobotomy scars on his forehead. Unwilling to allow McMurphy to live in such a state, the Chief smothers McMurphy to death with his pillow. He then carries out McMurphy's escape plan by lifting the hydrotherapy console off the floor and hurling the massive fixture through a grated window. Chief climbs through the window and runs off into the distance, with Taber waking up just in time to see him escape and cheering as the others awake.

Question: What does McMurphy offer Chief?
Answer: A piece of gum