Endowing robot policies with anticipatory capabilities has proven effective in improving control performance. However, existing methods predominantly rely on explicit supervision derived from future observations, in either pixel or latent space, at the cost of additional architectural and computational complexity. Departing from explicit future supervision, we ask whether such foresight can be elicited directly from the action signal. To this end, we propose \textbf{AdaPolicy}, an \textbf{A}ction-\textbf{d}riven \textbf{a}nticipatory policy framework that shapes future-aware representations via cascaded next-chunk prediction. Specifically, an Online Action Head jointly produces the current action chunk and an evolved latent state. Building on this state, a cascaded Unrolled Action Head forecasts the next chunk, and the resulting loss flows back through the evolved state to instill implicit foresight into the online policy. Importantly, this training-only unroll requires no future-observation encoding, keeping AdaPolicy lightweight to train with zero inference-time overhead. Experiments on MetaWorld, RoboTwin 2.0, LIBERO, and real-world deployments demonstrate consistent gains across diverse settings, including up to $14$ percentage points on millimeter-precision tasks such as earphone assembly and shoelace threading. More details and visualizations are available at \href{https://teee111.github.io/AdaPolicy/}{https://teee111.github.io/AdaPolicy/}.

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