Some context: The Osmonds are an American family music group with a long and varied career. The Osmond Brothers began as a barbershop quartet consisting of brothers Alan, Wayne, Merrill and Jay Osmond. They were later joined by younger siblings Donny and Jimmy, both of whom enjoyed success as solo artists as well. With the addition of Donny, the group became known as The Osmonds, and enjoyed its greatest success as both a boy band and a hard rock ensemble.
The Osmond Brothers' career began in 1958 when Alan, Wayne, Merrill and Jay began singing barbershop music for local audiences in and around Ogden. In their made-for-TV movie Inside the Osmonds, they explain that they originally performed to earn money to support Virl and Tom in buying hearing aids and serving missions for the church. Despite their young ages (Alan was 9, Wayne 7, Merrill 5, and Jay 3) and within a few years, the boys' talent and stage presence were strong enough that their father, George Osmond, took them to audition for Lawrence Welk in California. Welk was unable to meet with them, but on the same trip, they visited Disneyland. Tommy Walker, Disneyland's Director of Entertainment and Customer Relations from 1955 to 1966, found the Osmond Brothers singing with The Dapper Dans on Main Street. Walker hired the Osmonds to perform on a segment of "Disneyland After Dark". This episode aired on 4/15/62 and 7/01/62.  While the Osmond Brothers were performing on a televised Disney special, Andy Williams's father saw them and was so impressed he told his son to book them for his television show. Andy did, and the Osmond Brothers were regulars on the show from 1962 to 1969, where they earned the nickname "one-take Osmonds" among staff due to their professionalism and tireless rehearsing. Donny soon joined them on the show, making the Osmond Brothers a 5-member group. Marie and Jimmy were also introduced on the show as the years went by. During this time, the Osmonds also toured Europe, performing with Sweden's most popular singer, Lars Lonndahl, and even releasing a single where they sang a Swedish version of "Two Dirty Little Hands" ("Fem smutsiga sma fingrar").  The Andy Williams Show ended its first run in 1967, after which the Osmond Brothers were signed to The Jerry Lewis Show, staying with that show until it was canceled (and, coincidentally, The Andy Williams Show returned for a second run) in 1969. but soon decided they wanted to perform popular music and shed their variety-show image. They wanted to become a rock and roll band. The change was a difficult one for their father, who was suspicious of rock and roll, but he was persuaded and the boys began performing as a pop band.  To this end, the Osmonds recorded a single, "Flower Music", for UNI records in 1967. They achieved only modest success at first, but they found fame in 1971.
What influenced them?
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Some context: Antoine Dominique "Fats" Domino Jr. (February 26, 1928 - October 24, 2017) was an American pianist and singer-songwriter. One of the pioneers of rock and roll music, Domino sold more than 65 million records. Between 1955 and 1960, he had eleven Top 10 hits. His humility and shyness may be one reason his contribution to the genre has been overlooked.
Domino moved to ABC-Paramount Records in 1963. The label dictated that he record in Nashville, Tennessee, rather than New Orleans. He was assigned a new producer (Felton Jarvis) and a new arranger (Bill Justis). Domino's long-term collaboration with the producer, arranger, and frequent co-writer Dave Bartholomew, who oversaw virtually all of his Imperial hits, was seemingly at an end. Jarvis and Justis changed the Domino sound somewhat, notably by adding the backing of a countrypolitan-style vocal chorus to most of his new recordings. He released 11 singles for ABC-Paramount, several which hit the Top 100 but just once entering the Top 40 ("Red Sails in the Sunset", 1963). By the end of 1964 the British Invasion had changed the tastes of the record-buying public, and Domino's chart run was over.  Despite the lack of chart success, Domino continued to record steadily until about 1970, leaving ABC-Paramount in mid-1965 and recording for Mercury Records, where he delivered a live album and two singles. A studio album was planned but stalled with just four tracks recorded . Dave Bartholomew's small Broadmoor label (reuniting with Bartholomew along the way), featured many contemporary Soul infused sides but an album was released overseas in 1971 to fulfill his Reprise Records contract. He shifted to that label after Broadmoor and had a Top 100 single, a cover of the Beatles' "Lady Madonna".  Domino appeared in the Monkees' television special 33 1/3  Revolutions per Monkee in 1969. He continued to be popular as a performer for several decades. He made a cameo appearance in Clint Eastwood's movie Any Which Way You Can, filmed in 1979 and released in 1980 singing the country song "Whiskey Heaven" which later became a minor hit. His life and career were showcased in Joe Lauro's 2015 documentary The Big Beat: Fats Domino and the Birth of Rock 'n' Roll.
Where did he go after he left Imperial?
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1970, leaving ABC-Paramount in mid-1965 and recording for Mercury Records,