Study of the reactions $e^+e^-\to\pi^+\pi^-\pi^0\pi^0\pi^0\gamma$ and $\pi^+\pi^-\pi^0\pi^0\eta\gamma$ at center-of-mass energies from threshold to 4.35 GeV using initial-state radiation.

We study the processes $e^+e^-\to\pi^+\pi^-\pi^0\pi^0\pi^0\gamma$ and $\pi^+\pi^-\pi^0\pi^0\eta\gamma$ in which an energetic photon is radiated from the initial state. The data were collected with the BaBar~ detector at SLAC. About 14\,000 and 4700 events, respectively, are selected from a data sample corresponding to an integrated luminosity of 469~$fb^{-1}$. The invariant mass of the hadronic final state defines the effective $e^+e^-$ center-of-mass energy. From the mass spectra, the first precise measurement of the $e^+e^-\to\pi^+\pi^-\pi^0\pi^0\pi^0$ cross section and the first measurement ever of the $e^+e^-\to\pi^+\pi^-\pi^0\pi^0\eta$ cross section are performed. The center-of-mass energies range from threshold to 4.35~GeV. The systematic uncertainty is typically between 10 and 13\%. The contributions from $\omega\pi^0\pi^0$, $\eta\pi^+\pi^-$, and other intermediate states are presented. We observe the $J/\psi$ and $\psi(2S)$ in most of these final states and measure the corresponding branching fractions, many of them for the first time.