In a classic video game, balloons travel from left to right along a linear path that passes through a circular playing area as shown. The size and position of the playing area and the position of the linear path change as players repeat the game. Players are given one minute to pop as many balloons as possible within the playing area by dropping arrows from the tip of a launcher that can be moved horizontally along the top of the screen. The game's software tracks each balloon using the point at the top of the balloon. A balloon is within the playing area if the top point lies either on or inside the circle. In the figure, the balloons labeled 2, 3, and 4 are within the playing area. Balloons cannot be popped before they enter or after they leave the playing area. Saachi opened the game and was assigned a circular playing area and a linear path. The graphs in the zy-plane of her assigned playing area and linear path have equations (2-1)² + y² = 100 and y=3z +7, respectively. What is the maximum z-value at which Saachi can place the tip of her launcher in order to pop a balloon along the linear path within the playing area? Enter your answer in the box.