Recent advances in large language models (LLMs) offer promising potential for automating formal methods. However, applying them to formal verification remains challenging due to the complexity of specification languages, the risk of hallucinated output, and the semantic gap between natural language and formal logic. We introduce PAT-Agent, an end-to-end framework for natural language autoformalization and formal model repair that combines the generative capabilities of LLMs with the rigor of formal verification to automate the construction of verifiable formal models. In PAT-Agent, a \emph{Planning LLM} first extracts key modeling elements and generates a detailed plan using semantic prompts, which then guides a \emph{Code Generation LLM} to synthesize syntactically correct and semantically faithful formal models. The resulting code is verified using \emph{the PAT model checker} against user-specified properties, and when violations occur, a \emph{Repair Loop} is triggered to iteratively correct the model using counterexamples. To improve flexibility, we built a web-based interface that enables users, particularly non-FM-experts, to describe, customize, and verify system behaviors through user-LLM interactions. Experimental results on 40 systems show that PAT-Agent consistently outperforms baselines, achieving high verification success with superior efficiency. The ablation studies confirm the importance of both planning and repair components, and the user study demonstrates that our interface is accessible and supports effective formal modeling, even for users with limited formal methods experience.