What motivated Reynie and Kate to spy on the Executives in the gym?

The motivation of Reynie and Kate to spy on the Executives in the gym was rooted in the hope that their findings would support their overall mission. At this point in the narrative, the characters are deeply invested in uncovering the truth about the Institute's manipulative operations. Their actions are driven by a sense of urgency to gather critical information that could expose the Executives and help them in their efforts to stop any nefarious plans. The potential consequences of their observations could directly impact their broader goal, thus making this motivation a key element in their decision-making process.

While curiosity about the Executives' activities is certainly present, and there is an element of gathering information about the Institute’s operations, the primary drive is to assist their cause, especially given the precarious situation they find themselves in. This necessity to "aid their mission" encapsulates the urgency and importance of their actions, allowing them to reinforce their commitment to the larger objectives set by Mr. Benedict.