A strongman Bambula can carry several weights at the same time, if their total weight does not exceed $200$ kg, and these weights are no more than three. On the way to work, he injured his finger and found that he could now carry no more than two weights (and still no more than $200$ kg). At what minimum $k$ is the statement true: any set of $100$ weights that Bambula could previously carry in $50$ runs, with a sore finger, he will be able to carry in no more than $k$ runs?