On not racing ahead of the person leading
The instruction is about staying behind him, not with him.
إِنَّمَا الإِمَامُ لِيُؤْتَمَّ بِهِ فَلاَ تَخْتَلِفُوا عَلَيْهِ فَإِذَا كَبَّرَ فَكَبِّرُوا وَإِذَا رَكَعَ فَارْكَعُوا وَإِذَا قَالَ سَمِعَ اللَّهُ لِمَنْ حَمِدَهُ . فَقُولُوا اللَّهُمَّ رَبَّنَا لَكَ الْحَمْدُ . وَإِذَا سَجَدَ فَاسْجُدُوا وَإِذَا صَلَّى جَالِسًا فَصَلُّوا جُلُوسًا أَجْمَعُونَ
The imam is appointed to be followed. When he says: Allah hears whoever praises Him, say: our Lord, to You is praise.
The full narration
Abu Huraira reported:The Messenger of Allah (ﷺ) said: The Imam is appointed, so that he should be followed, so don't be at variance with him. Recite takbir when he recites it; bow down when he bows down and when he says:" Allah listens to him who praises Him," say:" O Allah, our Lord, to Thee be the Praise." And when he (the Imam) prostrates, you should also prostrate, and when he says prayer sitting, you should all observe prayer sitting
Source
Ṣaḥīḥ Muslim · 930
- Agreed upon as authentic (al-Bukhārī / Muslim)