  In the early 2000's, Baues computed the secondary Steenrod algebra, the
algebra of all secondary cohomology operations. Together with Jibladze, they
showed that this gives an algorithm that computes all Adams $d_2$ differentials
for the sphere.
  The goal of this paper is to reinterpret their results in the language of
synthetic spectra in order to achieve stronger computational results. Using
this, we obtain an algorithm that computes hidden extensions on the $E_3$ page
that jump by one filtration, in addition to the $d_2$ differentials of
Baues--Jibladze. We then implement and run this algorithm for the sphere up to
the 140th stem.
  Combined with a generalized version of the Leibniz rule, these hidden
extensions allow us to compute many longer differentials with ease. In
particular, we resolve all remaining unknown $d_2$, $d_3$, $d_4$ and $d_5$
differentials of the sphere up to the 95th stem.
