  By introducing simple topological constraints and applying a binary
decomposition method, we show the existence of a set of prograde double-double
orbits for any rotation angle $\theta \in (0, \pi/7]$ in the equal-mass
four-body problem.
  A new geometric argument is introduced to show that for any $\theta \in (0,
\pi/2)$, the action of the minimizer corresponding to the prograde
double-double orbit is strictly greater than the action of the minimizer
corresponding to the retrograde double-double orbit. This geometric argument
can also be applied to study orbits in the planar three-body problem, such as
the retrograde orbits, the prograde orbits, the Schubart orbit and the
H\'{e}non orbit.
