Using $448.0 times 10^6$~$psi(3686)$ events collected with the BESIII detector, an amplitude analysis is performed for $psi(3686)togammachi_{c1}$, $chi_{c1}toetapi^+pi^-$ decays. The most dominant two-body structure observed is $a_0(980)^{pm}pi^{mp}$; $a_0(980)^{pm}toetapi^{pm}$. The $a_0(980)$ line shape is modeled using a dispersion relation, and a significant non-zero $a_0(980)$ coupling to the $eta^{prime}pi$ channel is measured. We observe $chi_{c1}to a_2(1700)pi$ production for the first time, with a significance larger than 17$sigma$. The production of mesons with exotic quantum numbers, $J^{PC}=1^{-+}$, is investigated, and upper limits for the branching fractions $chi_{c1}to pi_1(1400)^{pm}pi^{mp}$, $chi_{c1}to pi_1(1600)^{pm}pi^{mp}$, and $chi_{c1}topi_1(2015)^{pm}pi^{mp}$, with subsequent $pi_1(X)^{pm} to etapi^{pm}$ decay, are determined.
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