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Remove meat from chicken, reserving skin and bones. Coarsely chop 1 celery rib and put in a 6- to 8-quart pot along with chicken bones and skin, onion, parsley sprigs, thyme sprigs, bay leaf, and chicken broth. Simmer, partially covered, 1 hour.
While broth is simmering, shred about half of chicken meat into 1-inch-long pieces (about 1/4 inch thick) to yield 1 1/2 cups meat, reserving remaining meat for another use. Cut remaining 2 celery ribs into 1/4-inch dice.
Pour chicken broth through a fine-mesh sieve into a large bowl, pressing hard on solids with back of a ladle and then discarding them. Skim fat from surface of broth.
Return strained broth to pot, then add carrots, diced celery, and rice and simmer, partially covered, stirring occasionally, until vegetables are tender and rice is very soft, about 30 minutes. Stir in shredded chicken and chopped parsley
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