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The multiple-access channel with feedback and correlated sources (MACFCS) models a sensor network in which sensors collect and transmit correlated data to a common sink. We present four achievable rate regions and a capacity outer bound for the MACFCS. For the first achievable region, we construct a decode-forward based coding strategy. The sources first exchange their data, and then cooperate to send full information to the destination. We term this strategy full decoding at sources with decode-forward (FDS-DF). For two of the other achievable regions, we first perform Slepian-Wolf coding to remove the correlation among the source data. This is followed by either (i) a compress-forward based coding strategy for the multiple-access channel with feedback, or (ii) an existing coding strategy for the multiple-access channel. We also find another achievable region using a multihop coding strategy, which only uses point-to-point coding (no cooperation). From numerical computations, we see that different strategies perform better under certain source correlation structures and network topologies. More specifically, FDS-DF approaches the capacity when (i) the inter-source distance decreases, or (ii) the correlation among the sources gets higher. Furthermore, the cooperative coding strategies considered support larger achievable rate regions than the noncooperative multihop strategy.