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We explore implications of a range of black hole (BH) seeding prescriptions on the formation of the brightest $z\gtrsim6$ quasars in cosmological hydrodynamic simulations. The underlying galaxy formation model is the same as in IllustrisTNG. Using constrained initial conditions, we study the growth of BHs in rare overdense regions (forming $\gtrsim10^{12}M_{\odot}/h$ halos by $z=7$) using a $(9~\mathrm{Mpc}/h)^3$ simulated volume. BH growth is maximal within halos that are compact and have a low tidal field. For these halos, we consider an array of gas-based seeding prescriptions wherein $M_{\mathrm{seed}}=10^4-10^6~M_{\odot}/h$ seeds are inserted in halos above critical thresholds for halo mass and dense, metal-poor gas mass (defined as $\tilde{M}_{\mathrm{h}}$ and $\tilde{M}_{\mathrm{sf,mp}}$, respectively, in units of $M_{\mathrm{seed}}$). We find that a seed model with $\tilde{M}_{\mathrm{sf,mp}}=5$ and $\tilde{M}_{\mathrm{h}}=3000$ successfully produces a $z\sim6$ quasar with $\sim10^9~M_{\odot}$ mass and $\sim10^{47}~\mathrm{ergs~s^ {-1}}$ luminosity. BH mergers play a crucial role at $z\gtrsim9$, causing an early boost in BH mass at a time when accretion-driven BH growth is negligible. When more stringent seeding conditions are applied (for e.g., $\tilde{M}_{\mathrm{sf,mp}}=1000$), the relative paucity of BH seeds results in a much lower merger rate. In this case, $z\gtrsim6$ quasars can only be formed if we enhance the maximum allowed BH accretion rates (by factors $\gtrsim10$) compared to the accretion model used in IllustrisTNG. This can be achieved either by allowing for super-Eddington accretion, or by reducing the radiative efficiency. Our results show that progenitors of $z\sim6$ quasars have distinct BH merger histories for different seeding models, which will be distinguishable with LISA observations.