Aspect-based sentiment analysis (ABSA) aims to identify aspects and their associated sentiments from text. It can be categorized into explicit sentiment analysis (ESA) and implicit sentiment analysis (ISA). Unlike ESA, ISA lacks explicit sentiment indicators, requiring deeper contextual understanding and multilevel reasoning to infer underlying opinions. To address this challenge, we propose a progressive chain-of-thought-based reasoning framework (P-CoR) inspired by advanced prompt engineering in large language models (LLMs). P-CoR consists of three progressive prompting strategies: chain-of-thought-based reasoning (CoR), chain self-consistency-based reasoning (CSCR), and tree-of-thought-based reasoning (ToR). In CoR, we adopt a three-step reasoning process to identify fine-grained aspects, infer aspect-level opinions, and predict sentiment polarity. ToR involves identifying all aspect terms and subsequently determining the intensity of sentiment and the underlying reasons for opinions associated with each aspect, while CSCR enhances robustness by selecting candidate answers through voting consistency across inferred aspects and opinions. Experimental results on two benchmark datasets show that all proposed strategies outperform traditional neural models, with ToR achieving the best overall performance. Specifically, ToR attains 86.12% accuracy and 79.22% Macro-F1 on the restaurant dataset and 83.09% accuracy with 83.90% Macro-F1 on the laptop dataset. These results highlight the importance of ISA and validate the effectiveness of progressive prompting strategies for sentiment inference using LLMs.