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We report the Transiting Exoplanet Survey Satellite (TESS) discovery of a three-planet system around the bright Sun-like star HD~22946(V=8.3 mag),also known as TIC~100990000, located 63 parsecs away.The system was observed by TESS in Sectors 3, 4, 30 and 31 and two planet candidates, labelled TESS Objects of Interest (TOIs) 411.01 (planet $c$) and 411.02 (planet $b$), were identified on orbits of 9.57 and 4.04 days, respectively. In this work, we validate the two planets and recover an additional single transit-like signal in the light curve, which suggests the presence of a third transiting planet with a longer period of about 46 days.We assess the veracity of the TESS transit signals and use follow-up imaging and time series photometry to rule out false positive scenarios, including unresolved binary systems, nearby eclipsing binaries or background/foreground stars contaminating the light curves. Parallax measurements from Gaia EDR3, together with broad-band photometry and spectroscopic follow-up by TFOP allowed us to constrain the stellar parameters of TOI-411, including its radius of$1.157\pm0.025R_\odot$. Adopting this value, we determined the radii for the three exoplanet candidates and found that planet $b$ is a super-Earth, with a radius of $1.72\pm0.10R_\oplus$, while planet $c$ and $d$ are sub-Neptunian planets, with radii of$2.74\pm0.14R_\oplus$ and $3.23\pm0.19R_\oplus$ respectively. By using dynamical simulations, we assessed the stability of the system and evaluated the possibility of the presence of other undetected, non-transiting planets by investigating its dynamical packing. We find that the system is dynamically stable and potentially unpacked, with enough space to host at least one more planet between $c$ and $d$.(Abridged), 21 pages, 12 figures. Accepted for publication on A&A