How do you escape the circle?

A dog runs continuously along the boundary of a circle of radius 1 at speed 4. You start at an arbitrary point strictly inside the circle and can run at speed 1 anywhere inside the disk.

If the dog is at the same boundary point at the instant you reach the boundary, you are caught. Otherwise, if you reach an unoccupied boundary point, you immediately escape.

Can you guarantee escape? If yes, describe a strategy and justify why it works.