merging figures
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@ -365,7 +365,6 @@ to standard WFT and $\Psi^\mu$ is the FCI wave function.
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%%% FIG 1 %%%
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\begin{figure*}
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\centering
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\includegraphics[width=0.7\linewidth]{algorithm.pdf}
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\caption{Algorithm showing the generation of the RS-DFT wave
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function $\Psi^{\mu}$ starting from $\Psi^{(0)}$.
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@ -459,7 +458,6 @@ stochastic reconfiguration algorithm developed by Assaraf \textit{et al.},
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\begin{table}
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\caption{FN-DMC energy $\EDMC$ (in \hartree{}) and number of determinants $\Ndet$ in \ce{H2O} for various trial wave functions $\Psi^{\mu}$ obtained with the srPBE density functional.}
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\label{tab:h2o-dmc}
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\centering
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\begin{ruledtabular}
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\begin{tabular}{crlrl}
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& \multicolumn{2}{c}{VDZ-BFD} & \multicolumn{2}{c}{VTZ-BFD} \\
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@ -485,7 +483,6 @@ stochastic reconfiguration algorithm developed by Assaraf \textit{et al.},
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%%% FIG 2 %%%
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\begin{figure}
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\centering
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\includegraphics[width=\columnwidth]{h2o-dmc.pdf}
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\caption{FN-DMC energy of \ce{H2O} as a function
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of $\mu$ for various levels of theory to generate
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@ -619,39 +616,29 @@ and solving Eq.~\eqref{eq:ci-j}.\cite{Nightingale_2001}
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We can easily compare $\Psi^\mu$ and $\Psi^J$ as they are developed
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on the same set of Slater determinants.
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In Fig.~\ref{fig:overlap}, we plot the overlaps
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$\braket*{\Psi^J}{\Psi^\mu}$ obtained for water as a function of $\mu$
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and, in Fig.~\ref{fig:dmc_small}, the FN-DMC energy of the wave function
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$\Psi^\mu$ as a function of $\mu$ together with that of $\Psi^J$.
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In Fig.~\ref{fig:overlap}, we plot the overlap
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$\braket*{\Psi^J}{\Psi^\mu}$ obtained for water as a function of $\mu$ (left graph)
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as well as the FN-DMC energy of the wave function
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$\Psi^\mu$ as a function of $\mu$ together with that of $\Psi^J$ (right graph).
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%%% FIG 3 %%%
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\begin{figure}
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\centering
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\begin{figure*}
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\includegraphics[width=\columnwidth]{overlap.pdf}
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\caption{Overlap between $\Psi^\mu$ and $\Psi^J$ as a function of $\mu$ for \ce{H2O}.
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\includegraphics[width=\columnwidth]{h2o-200-dmc.pdf}
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\caption{Left: Overlap between $\Psi^\mu$ and $\Psi^J$ as a function of $\mu$ for \ce{H2O}.
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Right: FN-DMC energy of $\Psi^\mu$ (red curve) as a function of $\mu$, together with
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the FN-DMC energy of $\Psi^J$ (blue line) for \ce{H2O}.
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The width of the lines represent the statistical error bars.
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For these two trial wave functions, the CI expansion consists of the 200 most important
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determinants of the FCI expansion obtained with the VDZ-BFD basis (see Sec.~\ref{sec:rsdft-j} for more details).}
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\label{fig:overlap}
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\end{figure}
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%%% %%% %%% %%%
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%%% FIG 4 %%%
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\begin{figure}
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\includegraphics[width=\columnwidth]{h2o-200-dmc.pdf}
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\caption{
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FN-DMC energies of $\Psi^\mu$ (red curve) as a function of $\mu$, together with
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the FN-DMC energy of $\Psi^J$ (blue line) for \ce{H2O}. The width of the lines
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represent the statistical error bars.
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For these two trial wave functions, the CI expansion consists of the 200 most important
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determinants of the FCI expansion obtained with the VDZ-BFD basis (see Sec.~\ref{sec:rsdft-j} for more details).}
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\label{fig:dmc_small}
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\end{figure}
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\end{figure*}
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%%% %%% %%% %%%
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As evidenced by Fig.~\ref{fig:overlap}, there is a clear maximum overlap between the two trial wave functions at $\mu=1$~bohr$^{-1}$, which
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coincides with the minimum of the FN-DMC energy of $\Psi^\mu$ (see Fig.~\ref{fig:dmc_small}).
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coincides with the minimum of the FN-DMC energy of $\Psi^\mu$.
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Also, it is interesting to notice that the FN-DMC energy of $\Psi^J$ is compatible
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with that of $\Psi^\mu$ for $0.5 < \mu < 1$~bohr$^{-1}$, as shown by the overlap between the red and blue bands in Fig.~\ref{fig:dmc_small}.
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with that of $\Psi^\mu$ for $0.5 < \mu < 1$~bohr$^{-1}$, as shown by the overlap between the red and blue bands.
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This confirms that introducing short-range correlation with DFT has
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an impact on the CI coefficients similar to a Jastrow factor.
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This is yet another key result of the present study.
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@ -660,7 +647,7 @@ This is yet another key result of the present study.
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\begin{table}
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\caption{\ce{H2O}, double-zeta basis set. Integrated on-top pair density $\expval{ P }$
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for $\Psi^J$ and $\Psi^\mu$ with different values of $\mu$.
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\titou{Please remove table and merge data in the Fig. 5.}}
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\titou{Please remove table and merge data in Fig. 5.}}
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\label{tab:table_on_top}
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\begin{ruledtabular}
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\begin{tabular}{cc}
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@ -680,28 +667,16 @@ This is yet another key result of the present study.
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\end{table}
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%%% %%% %%% %%%
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%%% FIG 5 %%%
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\begin{figure}
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%%% FIG 4 %%%
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\begin{figure*}
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\includegraphics[width=\columnwidth]{density-mu.pdf}
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\caption{One-electron density $n(\br)$ along
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the \ce{O-H} axis of \ce{H2O} as a function of $\mu$ for $\Psi^J$ (dashed curve) and $\Psi^\mu$.
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For these two trial wave functions, the CI expansion consists of the 200 most important
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determinants of the FCI expansion obtained with the VDZ-BFD basis (see Sec.~\ref{sec:rsdft-j} for more details).}
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\label{fig:n1}
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\end{figure}
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%%% %%% %%% %%%
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%%% FIG 6 %%%
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\begin{figure}
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\includegraphics[width=\columnwidth]{on-top-mu.pdf}
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\caption{On-top pair
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density $n_2(\br,\br)$ along the \ce{O-H} axis of \ce{H2O} as a function of $\mu$
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for $\Psi^J$ (dashed curve) and $\Psi^\mu$.
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\caption{One-electron density $n(\br)$ (left) and on-top pair
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density $n_2(\br,\br)$ (right) along the \ce{O-H} axis of \ce{H2O} as a function of $\mu$ for $\Psi^J$ (dashed curve) and $\Psi^\mu$.
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For these two trial wave functions, the CI expansion consists of the 200 most important
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determinants of the FCI expansion obtained with the VDZ-BFD basis (see Sec.~\ref{sec:rsdft-j} for more details).}
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\label{fig:n2}
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\end{figure}
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\label{fig:densities}
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\end{figure*}
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%%% %%% %%% %%%
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In order to refine the comparison between $\Psi^\mu$ and $\Psi^J$, we
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@ -714,9 +689,9 @@ report in Table~\ref{tab:table_on_top} the integrated on-top pair density
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where $n_2(\br_1,\br_2)$ is the two-body density [normalized to $\Nelec(\Nelec-1)$]
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obtained for both $\Psi^\mu$ and $\Psi^J$.
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Then, in order to have a pictorial representation of both the on-top
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pair density and the density, we report in Figs.~\ref{fig:n1} and \ref{fig:n2}
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pair density and the density, we report in Fig.~\ref{fig:densities}
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the plots of the total density $n(\br)$ and on-top pair density
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$n_2(\br,\br)$ along one of these \ce{O-H} axis of the water molecule.
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$n_2(\br,\br)$ along one of the \ce{O-H} axis of the water molecule.
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From these data, one can clearly notice several trends.
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First, from Table~\ref{tab:table_on_top}, we can observe that the overall
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@ -733,7 +708,7 @@ $\Psi^{\mu=0.5}$, and at a large distance the on-top pair density is
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the closest to $\mu=\infty$. The integrated on-top pair density
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obtained with $\Psi^J$ lies between the values obtained with
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$\mu=0.5$ and $\mu=1$~bohr$^{-1}$, consistently with the FN-DMC energies
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and the overlap curve depicted in Figs.~\ref{fig:overlap} and \ref{fig:dmc_small}
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and the overlap curve depicted in Fig.~\ref{fig:overlap}.
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These data suggest that the wave functions $\Psi^{0.5 \le \mu \le 1}$ and $\Psi^J$ are close,
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and therefore that the operators that produced these wave functions (\ie, $H^\mu[n]$ and $e^{-J}He^J$) contain similar physics.
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@ -747,7 +722,7 @@ increases the probability to find electrons at short distances in $\Psi^\mu$,
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while the effective one-body potential $\hat{\bar{V}}_{\text{Hxc}}^{\text{sr},\mu}[n_{\Psi^{\mu}}]$,
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provided that it is exact, maintains the exact one-body density.
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This is clearly what has been observed from
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Figs.~\ref{fig:n1} and \ref{fig:n2}.
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Fig.~\ref{fig:densities}.
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Regarding now the transcorrelated Hamiltonian $e^{-J}He^J$, as pointed out by Ten-no,\cite{Ten-no2000Nov}
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the effective two-body interaction induced by the presence of a Jastrow factor
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can be non-divergent when a proper Jastrow factor is chosen.
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@ -1075,7 +1050,7 @@ extrapolated FCI energies. The same comment applies to
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$\mu=0.5$~bohr$^{-1}$ with the quadruple-zeta basis set.
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%%% FIG 7 %%%
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%%% FIG 5 %%%
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\begin{figure*}
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\centering
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\includegraphics[width=\textwidth]{g2-dmc.pdf}
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@ -1101,7 +1076,7 @@ $\mu$. Although the FN-DMC energies are higher, the numbers show that
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they are more consistent from one system to another, giving improved
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cancellations of errors.
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%%% FIG 8 %%%
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%%% FIG 6 %%%
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\begin{figure*}
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\centering
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\includegraphics[width=\textwidth]{g2-ndet.pdf}
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