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We present the deepest constraints yet on the median rest-UV+optical SED of $z\sim10$ galaxies, prior to JWST science operations. We constructed stacks based on four robust $J_{125}$-dropouts, previously identified across the GOODS fields. We used archival HST/WFC3 data and the full depth Spitzer/IRAC mosaics from the GREATS program, the deepest coverage at $\sim3-5μ$m to date. The most remarkable feature of the SED is a blue IRAC $[3.6]-[4.5]=-0.18\pm0.25$ mag color. We also find a nearly flat $H_{160}-[3.6]=0.07\pm0.22$ mag color, corresponding to a UV slope $β= -1.92\pm0.25$. This is consistent with previous studies, and indicative of minimal dust absorption. The observed blue IRAC color and SED fitting suggest that $z\sim10$ galaxies have very young (few $\times10$ Myr) stellar populations, with $80\%$ of stars being formed in the last $\lesssim 160$ Myr ($2σ$). While an exciting result, the uncertainties on the SED are too large to allow us to place strong constraints on the presence of a nebular continuum in $z\sim10$ galaxies (as might be suggested by the blue $[3.6]-[4.5] < 0$ mag color). The resulting sSFR is consistent with the specific accretion rate of dark matter halos, indicative of a star-formation efficiency showing quite limited evolution at such early epochs.