We used the Atacama Large Millimeter/submillimeter Array (ALMA) to map $^{12}$CO($J$ = 1-0), $^{12}$CO($J$ = 2-1), $^{12}$CO($J$ = 3-2), $^{13}$CO($J$ = 2-1), and [CI]($^3P_1$-$^3P_0$) emission lines around the type 1 active galactic nucleus (AGN) of NGC 7469 ($z = 0.0164$) at $sim 100$ pc resolutions. The CO lines are bright both in the circumnuclear disk (central $sim 300$ pc) and the surrounding starburst (SB) ring ($sim 1$ kpc diameter), with two bright peaks on either side of the AGN. By contrast, the [CI]($^3P_1$-$^3P_0$) line is strongly peaked on the AGN. Consequently, the brightness temperature ratio of [CI]($^3P_1$-$^3P_0$) to $^{13}$CO(2-1) is $sim 20$ at the AGN, as compared to $sim 2$ in the SB ring. Our local thermodynamic equilibrium (LTE) and non-LTE models indicate that the enhanced line ratios (or CI enhancement) are due to an elevated C$^0$/CO abundance ratio ($sim 3-10$) and temperature ($sim 100-500$ K) around the AGN as compared to the SB ring (abundance ratio $sim 1$, temperature $lesssim 100$ K), which accords with the picture of the X-ray-dominated Region (XDR). Based on dynamical modelings, we also provide CO(1-0)-to- and [CI]($^3P_1$-$^3P_0$)-to-molecular mass conversion factors at the central $sim 100$ pc of this AGN as $alpha_{rm CO} = 4.1$ and $alpha_{rm CI} = 4.4~M_odot$ (K km s$^{-1}$ pc$^2$)$^{-1}$, respectively. Our results suggest that the CI enhancement is potentially a good marker of AGNs that could be used in a new submillimeter diagnostic method toward dusty environments.
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