The shape of a dog kennel from above is an equilateral triangle with side length $1$ m and its corners in points $A, B$ and $C$, as shown in the picture. The chain of the dog is of length $6$ m and its end is fixed to the corner in point $A$. The dog himself is in point $K$ in a way that the chain is tight and points $K, A$ and $B$ are on the same straight line. The dog starts to move clockwise around the kennel, holding the chain tight all the time. How long is the walk of the dog until the moment when the chain is tied round the kennel at full?