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Despite extensive efforts, discovery of high-energy astrophysical neutrino sources remains elusive. We present an event-level simultaneous maximum likelihood analysis of tracks and cascades using IceCube data collected from April 6, 2008 to May 23, 2022 to search the whole sky for neutrino sources and, using a source catalog, for coincidence of neutrino emission with gamma-ray emission. This is the first time a simultaneous fit of different detection channels is used to conduct a time-integrated all-sky scan with IceCube. Combining all-sky tracks, with superior pointing-power and sensitivity in the northern sky, with all-sky cascades, with good energy-resolution and sensitivity in the southern sky, we have developed the most sensitive point-source search to date by IceCube which targets the entire sky. The most significant point in the northern sky aligns with NGC 1068, a Seyfert II galaxy, which shows a 3.5$\sigma$ excess over background after accounting for trials. The most significant point in the southern sky does not align with any source in the catalog and is not significant after accounting for trials. A search for the single most significant Gaussian flare at the locations of NGC 1068, PKS 1424+240, and the southern highest significance point shows results consistent with expectations for steady emission. Notably, this is the first time that a flare shorter than four years has been excluded as being responsible for NGC 1068's emergence as a neutrino source. Our results show that combining tracks and cascades when conducting neutrino source searches improves sensitivity and can lead to new discoveries.