This paper presents a search for new heavy particles decaying into a pair of top quarks using 139 fb$^{-1}$ of proton--proton collision data recorded at a centre-of-mass energy of $\sqrt{s}=13$ TeV with the ATLAS detector at the Large Hadron Collider. The search is performed using events consistent with pair production of high-transverse-momentum top quarks and their subsequent decays into the fully hadronic final states. The analysis is optimized for resonances decaying into a $t\bar{t}$ pair with mass above 1.4 TeV, exploiting a dedicated multivariate technique with jet substructure to identify hadronically decaying top quarks using large-radius jets and evaluating the background expectation from data. No significant deviation from the background prediction is observed. Limits are set on the production cross-section times branching fraction for the new $Z'$ boson in a topcolor-assisted-technicolor model. The $Z'$ boson masses below 3.9 and 4.7 TeV are excluded at 95% confidence level for the decay widths of 1% and 3%, respectively.
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