We present a large spectroscopic campaign with Keck/MOSFIRE targeting Lyman-alpha emission (Ly$alpha$) from intrinsically faint Lyman-break Galaxies (LBGs) behind 12 efficient galaxy cluster lenses. Gravitational lensing allows us to probe the more abundant faint galaxy population to sensitive Ly$alpha$ equivalent width limits. During the campaign we targeted 70 LBG candidates with MOSFIRE Y-band, selected photometrically to cover Ly$alpha$ over the range $7<z<8.2$. We detect $S/N>5$ emission lines in 2 of these galaxies and find that they are likely Ly$alpha$ at $z=7.148pm0.001$ and $z=7.161pm0.001$. We present new lens models for 4 of the galaxy clusters, using our previously published lens models for the remaining clusters to determine the magnification factors for the source galaxies. Using a Bayesian framework that employs large scale reionization simulations of the intergalactic medium (IGM) as well as realistic properties of the interstellar medium and circumgalactic medium, we infer the volume-averaged neutral hydrogen fraction, $overline{x}_{mathrm{HI}}$, in the IGM during reionization to be $overline{x}_{mathrm{HI}}=0.88^{+0.05}_{-0.10}$ at $z=7.6pm0.6$. Our result is consistent with a late and rapid reionization scenario inferred by Planck.
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