Gustav is studying to become an interpreter. In this line of work, knowing several languages is a must, and Gustav is already fluent in French, Mandarin, Nahuatl and even Finnish. But there is one language that Gustav struggles with: Norwegian. Before Gustav can graduate, he must complete the infamous Norwegian Conclusive Proficiency Check.

This exam consists of $$$n$$$ yes/no questions. Answering correctly gives $$$+1$$$ point, answering incorrectly gives $$$-1$$$ point, and not answering gives $$$0$$$ points. To pass the exam, at least $$$k$$$ points are needed. For each question, Gustav has a guess that he knows is correct with probability $$$p_i$$$ ($$$0.5 \leq p_i \leq 1$$$). Note that $$$p_i \geq 0.5$$$, because otherwise guessing the opposite would be better.

What is the maximum possible probability of passing the exam, assuming we carefully choose which questions to answer and which ones to leave unanswered? Note that unlike a programming contest, the exam does not have instant feedback. So Gustav will answer the questions, hand in his answers, and only later be told the results.