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The rapid rise of computation- & data-intensive applications, such as machine learning training and inference, poses significant challenges for joint communication, computation, and caching (3C) scheduling over the network. In this work, we study a cache-enabled computation network, where each node can forward packets, cache data, and compute tasks. Each task produces non-cacheable results and requires two types of input data: (i) cacheable data orchestrated by the cloud server and (ii) non-cacheable, user-specific data uploaded together with the request from the user. Existing joint 3C scheduling approaches either assume restrictive network topologies or neglect the size of user-specific input data, resulting in the widely used symmetric routing assumption, where computation results always return to the user following the reverse path of the corresponding requests. We propose our joint 3C framework, MICS, a distributed framework that incorporates user-specific data size into the model and jointly schedules network operations to minimize convex flow and computation costs over arbitrary network topologies. By breaking the symmetric routing, MICS significantly reduces system costs. In MICS, a dual subgradient-based control plane ensures ergodic convergence of the 3C variables under the Slater condition, while the data plane executes practical 3C operations. Simulations demonstrate up to a 32.1% improvement in average request satisfaction time compared to baseline methods.