Family caregivers of individuals with dementia often experience significant emotional and cognitive burden, which may manifest in the language they use on social media. Identifying such posts is valuable for understanding caregiver needs and advancing mental health research. Task 3 of the SMM4H 2025 shared task focuses on classifying whether a tweet indicates that the user has a family member with dementia. The task requires sensitivity to both direct and indirect expressions of caregiving. We addressed this task using a prompt-based zero-shot classification system powered by large language models (LLMs). Our method leverages instruction-tuned models, including DeepSeek-R1 and Mixtral-8x7B. To further evaluate our results, we developed a LLM-based multi-agent system to analyze cognitive distortions in tweets labeled as caregiver-related. The resulting distortion patterns offer psychological insight into the model's predictions and highlight the system's potential for broader applications in mental health monitoring.

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