Large Language Models (LLM)-based Multi-Agent Systems (MASs) have emerged as a new paradigm in software system design, increasingly demonstrating strong reasoning and collaboration capabilities. As these systems become more complex and autonomous, effective failure management is essential to ensure reliability and availability. However, existing approaches often rely on per-trace reasoning, which leads to low efficiency, and neglect historical failure patterns, limiting diagnostic accuracy. In this paper, we conduct a preliminary empirical study to demonstrate the necessity, potential, and challenges of leveraging historical failure patterns to enhance failure management in MASs. Building on this insight, we propose \textbf{EAGER}, an efficient failure management framework for multi-agent systems based on reasoning trace representation. EAGER employs unsupervised reasoning-scoped contrastive learning to encode both intra-agent reasoning and inter-agent coordination, enabling real-time step-wise failure detection, diagnosis, and reflexive mitigation guided by historical failure knowledge. Preliminary evaluations on three open-source MASs demonstrate the effectiveness of EAGER and highlight promising directions for future research in reliable multi-agent system operations.