A duck, pursued by a fox, escapes to the center of a perfectly circular pond. The fox cannot swim, and the duck cannot take flight from the water. The fox is $k$ times faster than the duck. Assuming the fox pursues an optimal strategy, how can the duck reach the edge of the pond and fly away without being eaten?
First, find a strategy by which the duck can escape if $k=4$. Then, determine the smallest value of $k$ such that it is impossible for the duck to escape.