Can $n$ circles on a plane generate $m$ intersection points where at least $k$ circles intersect?  

For $k = 2$ the answer is obvious since we can always place circles so that every one of them intersects every other, generating in total at most $2 {n\choose{2}}$ intersection points of $2$ circles.

What can we say for $k = 3$? 
In particular I am interested in  $n = 7$, $m = 12$. 

It is known (see figure) that $8$ circles can generate $12$ intersection points of at least $3$ circles. 

           
![enter image description here][1]

The question is: can we generate $12$ intersection points of at least $3$ circles using only $7$ circles in total?


  [1]: https://i.sstatic.net/qwyGLm.png