What type of plant has slender, red, freely branched stems with filiform leaves arranged like fans at the end of the stems?

The correct identification of the plant described in the question focuses on the characteristics of sago pondweed. Sago pondweed is known for its slender, reddish stems that can freely branch. The filiform leaves, which are long and slender, often cluster at the ends of these stems, resembling a fan-like arrangement. This distinctive appearance helps you identify sago pondweed in aquatic environments.

Other options, such as brittle naiad, coontail, and wild celery, have different physical characteristics. Brittle naiad, for instance, has a more rigid stem structure and different leaf arrangement. Coontail features a more bushy appearance with distinctive whorled leaves, and wild celery has broader, more robust stems and longer, wider leaves that do not have the fan-like configuration described. Recognizing these unique traits is essential for distinguishing various aquatic plants effectively.