Large language models (LLMs) have shown impressive multilingual capabilities through pretraining on diverse corpora. Although these models show strong reasoning abilities, their performance varies significantly between languages due to the imbalanced distribution of training data. Existing approaches using sample-level translation for extensive multilingual pretraining and cross-lingual tuning face scalability challenges and often fail to capture nuanced reasoning processes across languages. In this paper, we introduce AdaMCOT (Adaptive Multilingual Chain-of-Thought), a framework that enhances multilingual factual reasoning by dynamically routing thought processes in intermediary "thinking languages" before generating target-language responses. AdaMCOT leverages a language-agnostic core and incorporates an adaptive, reward-based mechanism for selecting optimal reasoning pathways without requiring additional pretraining. Our comprehensive evaluation across multiple benchmarks demonstrates substantial improvements in both factual reasoning quality and cross-lingual consistency, with particularly strong performance gains in low-resource language settings. An in-depth analysis of the model's hidden states and semantic space further elucidates the underlying mechanism of our method. The results suggest that adaptive reasoning paths can effectively bridge the performance gap between high and low-resource languages while maintaining cultural and linguistic nuances.