A search is conducted for a new scalar boson $S$, with a mass distinct from that of the Higgs boson, decaying into four leptons ($\ell =$$e$, $\mu$) via an intermediate state containing two on-shell, promptly decaying new spin-1 bosons $Z_\text{d}$: $S \rightarrow Z_\text{d}Z_\text{d} \rightarrow 4\ell$, where the $Z_\text{d}$ boson has a mass between 15 and 300 GeV, and the $S$ boson has a mass between either 30 and 115 GeV or 130 and 800 GeV. The search uses proton-proton collision data collected with the ATLAS detector at the Large Hadron Collider with an integrated luminosity of 139 fb$^{-1}$ at a centre-of-mass energy of $\sqrt{s}=13$ TeV. No significant excess above the Standard Model background expectation is observed. Upper limits at 95% confidence level are set on the production cross-section times branching ratio, $\sigma(gg \to S) \times \mathcal{B}(S\rightarrow Z_\text{d}Z_\text{d} \rightarrow 4\ell)$, as a function of the mass of both particles, $m_S$ and $m_{Z\text{d}}$.
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