Starting from the point (0,0), a grasshopper makes a series of moves on the coordinate plane. the grasshopper's first move takes her to (3,4). she then proceeds via this rule:
move rule: after a move which adds (x, y) to the grasshopper's coordinates, her next move adds either (x, y-1) or (x-1,y) to her coordinates.
thus, her second move adds either (3,3) or (2,4) to her position, landing her at (6,7) or (5,8) accordingly.
if the grasshopper's x and y coordinates never decrease, and if she ends at the point (a, a) where a is as large as possible, then what is the value of a?