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The intense photon fluxes from relativistic nuclei provide an opportunity to study photonuclear interactions in ultraperipheral collisions. The measurement of coherently photoproduced π+π−π+π− final states in ultraperipheral Pb-Pb collisions at sNN−−−√=5.02 TeV is presented for the first time. The cross section, dσ/dy, times the branching ratio (ρ→π+π+π−π−) is found to be 47.8±2.3 (stat.)±7.7 (syst.) mb in the rapidity interval |y|<0.5. The invariant mass distribution is not well described with a single Breit-Wigner resonance. The production of two interfering resonances, ρ(1450) and ρ(1700), provides a good description of the data. The values of the masses (m) and widths (Γ) of the resonances extracted from the fit are m1=1385±14 (stat.)±3 (syst.) MeV/c2, Γ1=431±36 (stat.)±82 (syst.) MeV/c2, m2=1663±13 (stat.)±22 (syst.) MeV/c2 and Γ2=357±31 (stat.)±49 (syst.) MeV/c2, respectively. The measured cross sections times the branching ratios are compared to recent theoretical predictions.