Split lines at given point

Good day , I have these 4 lines. Each line is an aggregation of two different lines (blue and yellow). I want to split these 4 lines into 2 lines each where they intersect with a given point (red dot). I used the intersector and the point on line overlayer. But in both cases I get 3 lines as a result instead of 2. These two transformers split the lines at point intersection as well as desaggregating the blue and yellow line part. This is not what I want. I only want to 2 lines (split at point intersection) as a result of the transformation. How can I achieve it ? I can not split lines at point intersection before I combine the lines. It’s a complex workbench.