The rise of large language models (LLMs) has revolutionized user interactions with knowledge-based systems, enabling chatbots to synthesize vast amounts of information and assist with complex, exploratory tasks. However, LLM-based chatbots often struggle to provide personalized support, particularly when users start with vague queries or lack sufficient contextual information. This paper introduces the Collaborative Assistant for Personalized Exploration (CARE), a system designed to enhance personalization in exploratory tasks by combining a multi-agent LLM framework with a structured user interface. CARE's interface consists of a Chat Panel, Solution Panel, and Needs Panel, enabling iterative query refinement and dynamic solution generation. The multi-agent framework collaborates to identify both explicit and implicit user needs, delivering tailored, actionable solutions. In a within-subject user study with 22 participants, CARE was consistently preferred over a baseline LLM chatbot, with users praising its ability to reduce cognitive load, inspire creativity, and provide more tailored solutions. Our findings highlight CARE's potential to transform LLM-based systems from passive information retrievers to proactive partners in personalized problem-solving and exploration.