Aims: $gamma$ rays can be used as a tracer in the search of sources of Galactic cosmic rays (CRs). We present deep observations of the Galactic Centre (GC) region with the MAGIC telescopes, which we use for inferring the underlying CR distribution. Methods: We observed the GC region for ${approx}100$ hours with the MAGIC telescopes from 2012 to 2017, at high zenith angles (58-70~deg). This implies a larger energy threshold, but also an increased effective collection area compared to low zenith observations. Using new software, we derive instrument response and background models, enabling us to study the diffuse emission in the region. We use pre-existing data of the gas distribution in the GC region to derive the underlying distribution of CRs. Results: We obtain a significant detection for all four model components used to fit our data (Sgr~A*, ``Arc, G0.9+0.1, and an extended component for the Galactic Ridge). We find that the diffuse component is best described as a power-law with index 2 and an exponential cut-off at around 20~TeV with the significance of the cut-off being only 2~$sigma$. The derived cosmic-ray profile hints to a peak at the GC position, with a measured profile index of $1.2 pm 0.3$, supporting the hypothesis of a CR accelerator at the GC. We argue that the measurements of this profile are presently limited by our knowledge of the gas distribution in the GC vicinity.
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