The number of the circles which are tangent to two circles and to a line

Suppose that we have two distinct circles and a line on a plane and that the distance between the centers of the circles is bigger than the sum of their radiuses. Also, suppose that the two circles exist in the same half plane divided by the line.

Question : If we consider a circle which is tangent both to the two circles and to the line, then what is the max of the number of such circles?

I have the following conjecture.

Conjecture : The max of the number of such circles is $8$.

We can see that $8$ is possible.

However, I'm facing difficulty in proving the conjecture (but it is just a conjecture). Can anyone help?

This is the Apollonius third problem with a degenerate circle.

The paragraph algebraic solutions of the Wikipedia article clearly shows that there are at most eight ($2^3$) solutions.