The Devil's Horizon question

I loved the whole book, was especially taken surprise by the ending. I have one question for the author, or anyone who can answer it:

(view spoiler)[At what point did Kate and Gabe become friendly again and decide to trick Frederick? Maybe I need to read it again but I saw no clues for this. (hide spoiler)]

Please put your answer in spoiler tags so as not to ruin the ending for anyone.