In clinical diagnosis, sequential screening, and cascaded prediction systems, information is often acquired in stages, with intermediate predictions guiding whether to act on the available evidence or defer action to gather further information. We study sequential predictive inference in these settings, aiming to construct a sequence of prediction sets that supports timely, resource-efficient decisions while remaining reliable throughout the acquisition process. Our objective is to maximize process-level utility subject to any-stage validity, which controls the probability of miscoverage anywhere in the prediction sequence. We develop COINS, a conformal framework for constructing such sequences, and establish a universality theorem identifying the common exclusion structure underlying all nested, calibration-symmetric, any-stage valid prediction sequences. Building on COINS, we develop Vopt-COINS for data-driven optimization of prediction sequences under task-specific utilities based on prediction-set size, stopping time and acquisition costs. Branchwise and localized extensions further support individualized acquisition and calibration. Numerical studies demonstrate the effectiveness and practical benefits of the proposed framework for reliable, resource-efficient sequential prediction.

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