The outflows of oxygen-rich asymptotic giant branch (AGB) stars are thought to be driven by radiation pressure by photon scattering on grains with sizes of tenths of microns. The details of the formation of dust in the extended atmospheres of these stars and the mass-loss process is still not well understood. We obtained quasi-simultaneous observations of the AGB star Mira using ALMA and ZIMPOL to probe the distribution of gas and large dust grains, respectively. The polarized light images show dust around Mira~A, the companion (Mira~B) and in a trail that connects the two sources. ALMA reveals that dust around Mira~A is contained in a high-gas-density region with a significant fraction of the polarized light arising from its edge. We constrained the gas density, temperature, and velocity within a few stellar radii from the star by modelling the CO $v=1, J=3-2$ line. We find a mass $(sim 3.8 pm 1.3) times 10^{-4}~M_odot$ to be contained between the stellar millimetre photosphere, $R^{rm 338~GHz}_star$, and $4~R^{rm 338~GHz}_star$. Our best-fit models with lower masses also reproduce the $^{13}$CO $v=0, J=3-2$ line emission from this region. We find TiO$_2$ and AlO abundances corresponding to 4.5% and $< 0.1$% of the total titanium and aluminium expected for a solar-composition gas. The low abundance of AlO allows for efficient Al depletion into dust already very close to the star, as expected from thermal dust emission observations and theoretical calculations of Mira variables. We constrain the presence of aluminium oxide grains based on the scattered light observations and our gas-phase model. We find that aluminium oxide grains can account for a significant fraction of the total aluminium atoms in this region only if the grains have sizes $lesssim 0.02~mu$m. This is an order of magnitude smaller than the maximum sizes predicted by dust-formation and wind-driving models.
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