Assume that $L_1$ and $L_2$ are Lagrangian submanifolds (of dimension at least 2) which intersect transversally. Do we always get a connected Lagrangian after performing Lagrangian surgeries at the points of intersections?

If not, what is an example?

Assume that $L_1$ and $L_2$ are connected Lagrangian submanifolds (of dimension at least 2) which intersect transversally. Do we always get a connected Lagrangian after performing Lagrangian surgeries at the points of intersections?

If not, what is an example?