What is the story about Guan Zhong’s departure from the state of Lu?

In the Spring and Autumn Period, Duke Huan of Qi, the king of Qi, had just ascended the throne. Under Bao Shuya's plan, he defeated the troops of Duke Zhuang of Lu, forced the State of Lu to kill the prince Jiu who was competing with him, and demanded that the State of Lu make peace with him. Guan Zhong, the teacher of Prince Jiu. Lu Zhuanggong of Lu State believed that Guan Zhong was a resourceful genius. If he was allowed to return to Qi State alive, once he was reused in the future, Qi State would definitely dominate the world, so he wanted to kill Guan Zhong. Xi Peng, the envoy of Qi State, said to Duke Zhuang of Lu: "Guan Zhong once shot and wounded Duke Huan of Qi. Duke Huan hated him so much that he wanted to kill him with his own hands to relieve his anger." Duke Zhuang of Lu believed it and then agreed. Let Guan Zhong go back alive, but put Guan Zhong in a prison car. Guan Zhong thought in the prison car that it must be the idea of ??his friend Bao Shuya to allow me to go back alive from the state of Lu. If they regret it later, they will send troops to chase me, so we must move forward quickly and leave Lu as soon as possible. In order to speed up the march, Guan Zhong composed a song called "Song of the Yellow Bird", which was melodious and exciting. He taught the servants pushing the prison cart to sing this song. The servants walked while singing, enjoying themselves and forgetting about fatigue. They walked more and more energetically, covering two days' journey in one day. Guan Zhong's worries were not unnecessary. After letting Guan Zhong go, Duke Zhuang of Lu regretted it and sent troops to pursue him again. However, it was too late. Guan Zhong and others had already entered the territory of Qi.