What is the greatest number of points of intersection that can occur when $2$ different circles and $2$ different straight lines are drawn on the same piece of paper?

What is the greatest number of points of intersection that can occur when different circles and different straight lines are drawn on the same piece of paper?

Solution 1: Make a diagram. Two geometric figures intersect if they have one or more points in common. Draw two circles which intersect in points. Draw a line which intersects the two circles in points. Draw another line which intersects the two circles in points and also intersects the first line. There are points of intersection.[asy]

draw(Circle((-0.7,0),1)); draw(Circle((0.7,0),1));

dot((0,0));

dot((0,0.7)); dot((0,-0.7));

draw((0,0)--(-2,0.6),Arrow); draw((0,0)--(-2,-0.6),Arrow); draw((0,0)--(2,0.6),Arrow); draw((0,0)--(2,-0.6),Arrow);

dot((-1.58,0.47)); dot((-1.58,-0.47)); dot((1.58,0.47)); dot((1.58,-0.47));

dot((-0.29,0.08)); dot((-0.29,-0.08)); dot((0.29,0.08)); dot((0.29,-0.08));

[/asy] Solution 2: Make a table of the maximum number of points of intersection.

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