Abstract While large language models (LLMs) are widely used across cultures, they often generate culturally inappropriate responses in unfamiliar cultural contexts due to biases embedded in their training data. Existing approaches primarily rely on expanding static cultural knowledge, which fails to capture the inherently relative and context-dependent nature of culture. In this paper, we propose a Cultural value-based Reasoning (CURE) framework that interprets behaviors through underlying cultural value systems. In addition, we integrate CURE into LLMs via Chain-of-Thought (CoT) distillation, referred to as CURE-distillation, to internalize culturally grounded reasoning. Experimental results show that models trained with CURE-distillation improve cultural adaptability, enabling them to produce culturally aligned ethical judgments across diverse cultural scenarios. These results suggest that strengthening sociocultural reasoning capabilities can substantially improve the cultural adaptability of LLMs. The code is available at https://github.com/KUNLP/CURE.