Cyber Threat Intelligence (CTI) reports are factual records compiled by security analysts through their observations of threat events or their own practical experience with attacks. In order to utilize CTI reports for attack detection, existing methods have attempted to map the content of reports onto system-level attack provenance graphs to clearly depict attack procedures. However, existing studies on constructing graphs from CTI reports suffer from problems such as weak natural language processing (NLP) capabilities, discrete and fragmented graphs, and insufficient attack semantic representation. Therefore, we propose a system called CRUcialG for the automated reconstruction of attack scenario graphs (ASGs) by CTI reports. First, we use NLP models to extract systematic attack knowledge from CTI reports to form preliminary ASGs. Then, we propose a four-phase attack rationality verification framework from the tactical phase with attack procedure to evaluate the reasonability of ASGs. Finally, we implement the relation repair and phase supplement of ASGs by adopting a serialized graph generation model. We collect a total of 10,607 CTI reports and generate 5,761 complete ASGs. Experimental results on CTI reports from 30 security vendors and DARPA show that the similarity of ASG reconstruction by CRUcialG can reach 84.54%. Compared with SOTA (EXTRACTOR and AttackG), the recall of CRUcialG (extraction of real attack events) can reach 88.13% and 94.46% respectively, which is 40% higher than SOTA on average. The F1-score of attack phase verification is able to reach 90.04%.