6. Along the diagonals of the bases $AC$ and $B_{1}D_{1}$ of the cube $ABCD A_{1}B_{1}C_{1}D_{1}$ with edge $a$, two ants, Gosha and Lesha, are crawling. They started moving simultaneously from points $A$ and $B_{1}$ respectively at a constant speed, with Lesha's speed being four times the speed of Gosha's movement, and they finished when Lesha reached point $D_{1}$. What is the minimum distance that separated Gosha and Lesha during their movement?
Answer: $d_{\text {min }}=a \sqrt{\frac{43}{34}}$