The astrophysical $s$-process is one of the two main processes forming elements heavier than iron. A key outstanding uncertainty surrounding $s$-process nucleosynthesis is the neutron flux generated by the ${}^{22}mathrm{Ne}(alpha, n){}^{25}mathrm{Mg}$ reaction during the He-core and C-shell burning phases of massive stars. This reaction, as well as the competing ${}^{22}mathrm{Ne}(alpha, gamma){}^{26}mathrm{Mg}$ reaction, is not well constrained in the important temperature regime from ${sim} 0.2$--$0.4$~GK, owing to uncertainties in the nuclear properties of resonances lying within the Gamow window. To address these uncertainties, we have performed a new measurement of the ${}^{22}mathrm{Ne}({}^{6}mathrm{Li}, d){}^{26}mathrm{Mg}$ reaction in inverse kinematics, detecting the outgoing deuterons and ${}^{25,26}mathrm{Mg}$ recoils in coincidence. We have established a new $n / gamma$ decay branching ratio of $1.14(26)$ for the key $E_x = 11.32$ MeV resonance in $^{26}mathrm{Mg}$, which results in a new $(alpha, n)$ strength for this resonance of $42(11)~mu$eV when combined with the well-established $(alpha, gamma)$ strength of this resonance. We have also determined new upper limits on the $alpha$ partial widths of neutron-unbound resonances at $E_x = 11.112,$ $11.163$, $11.169$, and $11.171$ MeV. Monte-Carlo calculations of the stellar ${}^{22}mathrm{Ne}(alpha, n){}^{25}mathrm{Mg}$ and ${}^{22}mathrm{Ne}(alpha, gamma){}^{26}mathrm{Mg}$ rates, which incorporate these results, indicate that both rates are substantially lower than previously thought in the temperature range from ${sim} 0.2$--$0.4$~GK.
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