進む and 進める: progress and moving work forward
Compare movement with work progressing, and distinguish 進める for moving work forward from 勧める for recommending.
Move forward
この道をまっすぐ進んでください。 directs someone straight along this road. 進む, read すすむ, has the te-form 進んで; 道を is the route.
工事のため、これ以上先には進めません。 says that construction prevents going any farther. Here 進めません negates the potential form of 進む.
Work progressing
準備が順調に進んでいます。 says that preparations are progressing smoothly. 進む can describe progress without movement between places.
相談しながら計画を進めています。 means that people are moving a plan forward while discussing it. With 進める, read すすめる, を marks the plan being advanced.
The spelling and meaning of すすめる
友達にこの本を勧めました。 says that someone recommended this book to a friend. Recommendation uses 勧める, unlike 進める for advancing work or procedures.
作業は進みましたが、まだ終わっていません。 says that the work has progressed but is not yet finished. Progress is not equivalent to completion.
Must 進めません in これ以上先には進めません mean not advancing a plan?
Show the answer and why
No. Here it is the potential form of 進む, meaning that one cannot go farther. Use the objects and movement context to interpret it.