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It was a rollercoaster few days for Beverly native Angie Miller on “American Idol.” On Saturday she enjoyed a triumphant return to her hometown. On Wednesday she gave three well-received performances buoyed by that trip home. And on Thursday the teen bid a tearful farewell when she was eliminated from the Fox singing competition. Buffeted by her fellow top three contestants Candice Glover and Kree Harrison — who advanced to next week’s finale — Miller watched a video montage of her journey from her first audition in New York City through the various stages of the competition. A cheerful presence on the show, judge Mariah Carey deemed Miller “festive” since she was often seen smiling and clapping her hands joyfully. Miller received high marks throughout season 12, with the show’s panel of judges routinely praising the quality and power of her voice, her piano playing, and original songwriting, with Carey calling her potential limitless.