Due to reasons not to be disclosed here, but related to the very same friend from the previous question, you have found yourself at a point p in a lake. Crocodiles are habituating the lake, strongly suggesting that youd swim to the nearest point on a the lakes shore before becoming a crocodile treat. As before, you have a GPS, but no map of the lake. Suggest your strategy, so the distance you have to swim is at most c d. Here d is the distance from your starting position p to the nearest point q of the shore of the lake, and c is a constant (not depending on d).

Note that the lake is not convex, but there are no island in the lake, so its shore is continuous. Show the best value of c. You do not have a map, and you dont know the crocodiles locations.

Note also that you do not have to find q itself. As usual, if your solution is iterative, describe what exactly happens at the ith iteration (for an arbitrary i).