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Fully convolutional neural networks have made promising progress in joint liver and liver tumor segmentation. Instead of following the debates over 2D versus 3D networks (for example, pursuing the balance between large-scale 2D pretraining and 3D context), in this paper, we novelly identify the wide variation in the ratio between intra- and inter-slice resolutions as a crucial obstacle to the performance. To tackle the mismatch between the intra- and inter-slice information, we propose a slice-aware 2.5D network that emphasizes extracting discriminative features utilizing not only in-plane semantics but also out-of-plane coherence for each separate slice. Specifically, we present a slice-wise multi-input multi-output architecture to instantiate such a design paradigm, which contains a Multi-Branch Decoder (MD) with a Slice-centric Attention Block (SAB) for learning slice-specific features and a Densely Connected Dice (DCD) loss to regularize the inter-slice predictions to be coherent and continuous. Based on the aforementioned innovations, we achieve state-of-the-art results on the MICCAI 2017 Liver Tumor Segmentation (LiTS) dataset. Besides, we also test our model on the ISBI 2019 Segmentation of THoracic Organs at Risk (SegTHOR) dataset, and the result proves the robustness and generalizability of the proposed method in other segmentation tasks.