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The processes hc P (P = ^, ~, ~⁰) are studied with a sample of (27. 120. 14) 10^8 (3686) events collected by the BESIII detector at the BEPCII collider. The decay h₂ is observed for the first time with the significance of 9. 0\, , and the branching fraction is determined to be (3. 770. 550. 130. 26) 10^-4, while B (h₂^) is measured to be (1. 400. 110. 040. 10) 10^-3, where the first uncertainties are statistical, the second systematic, and the third from the branching fraction of (3686) ^0hc. The combination of these results allows for a precise determination of R₇₂=B (hc) B (hc^), which is calculated to be (27. 04. 41. 0) \%. The results are valuable for gaining a deeper understanding of -^ mixing, and its manifestation within quantum chromodynamics. No significant signal is found for the decay hc^0, and an upper limit is placed on its branching fraction of B (hc^0) <5. 010^-5, at the 90\% confidence level.
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