Hamiltonian $S^1$-manifolds which are like a coadjoint orbit of $G_2$

Consider a compact symplectic manifold of dimension $2n$ with a Hamiltionan circle action. Then there are at least $n+1$ fixed points. Motivated by recent works on the case that the fixed point set consists of precisely $n+1$ isolated points, this paper studies a Hamiltonian $S^1$ action on a $10$-dimensional compact symplectic manifold with exactly 6 isolated fixed points. We study the relations of the following data: the first Chern class of the manifold,the largest weight of the action, all the weights of the action,the total Chern class of the manifold, and the integral cohomology ring of the manifold. We show how certain data can determine the others and show the similarities of these data with those of a coadjoint orbit of the exceptional Lie group $G_2$, equipped with a Hamiltonian action of a subcircle of the maximal torus.