The integration of Socratic questioning with chain-of-thought prompting emerges as a key strategy, promoting deeper engagement and rigorous inquiry into multifaceted challenges. Additionally, the mixed-method approaches presented in this work demonstrate the flexibility of Socratic prompting in addressing diverse research problems, bridging theoretical understanding with practical applications. While CoT prompting focuses on first-order interactions-evaluating LLM outputs and selecting the appropriate Socratic principle-the mixed approach extends beyond individual prompts, examining the higher-order dynamics between sequential prompts in a structured reasoning process. Together, these approaches enhance the precision, adaptability, and creativity of LLM outputs, positioning this framework as a critical tool for solving modern scientific questions.