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Double-source-plane strong gravitational lenses (DSPLs), with two sources at different redshifts, are independent cosmological probes of the dark energy equation of state parameterwand the matter density parameter Ω_m. We present the lens model for the DSPL AGEL035346−170639 and infer cosmological constraints from this system for flat Λ cold dark matter and flatwCDM cosmologies. From the joint posterior ofwand Ω_min the flatwCDM cosmology, we extract the following median values and 1σuncertainties: $w=-1.52_{-0.33}^{+0.49}$and ${\mathrm{\Omega }}_{\mathrm{m}}=0.192_{-0.131}^{+0.305}$from AGEL0353 alone. Combining our measurements with two previously analyzed DSPLs, we present the joint constraint on these parameters from a sample of three, the largest galaxy-scale DSPL sample used for cosmological measurement to date. The combined precision ofwfrom three DSPLs is higher by 15% over AGEL0353 alone. Combining DSPL and cosmic microwave background (CMB) measurements improves the precision ofwfrom CMB-only constraints by 39%, demonstrating the complementarity of DSPLs with the CMB. Despite their promising constraining power, DSPLs are limited by sample size, with only a handful discovered so far. Although ongoing and near-future wide-area sky surveys will increase the number of known DSPLs by up to two orders of magnitude, these systems will still require dedicated high-resolution imaging and spectroscopic follow-ups like those presented in this paper. Our ASTRO 3D Galaxy Evolution with Lenses collaboration is undertaking such follow-up campaigns for several newly discovered DSPLs and will provide cosmological measurements from larger samples of DSPLs in the future.