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Reflective listening is a fundamental skill that counselors must acquire to achieve proficiency in motivational interviewing (MI). It involves responding in a manner that acknowledges and explores the meaning of what the client has expressed in the conversation. In this work, we introduce the task of counseling response rewriting, which transforms non-reflective statements into reflective responses. We introduce VERVE, a template-based rewriting system with paraphrase-augmented training and adaptive template updating. VERVE first creates a template by identifying and filtering out tokens that are not relevant to reflections and constructs a reflective response using the template. Paraphrase-augmented training allows the model to learn less-strict fillings of masked spans, and adaptive template updating helps discover effective templates for rewriting without significantly removing the original content. Using both automatic and human evaluations, we compare our method against text rewriting baselines and show that our framework is effective in turning non-reflective statements into more reflective responses while achieving a good content preservation-reflection style trade-off.