Circle actions on oriented manifolds with 3 fixed points

Let the circle group act on a compact oriented manifold \(M\) with a non-empty discrete fixed point set. Then the dimension of \(M\) is even. If \(M\) has one fixed point, \(M\) is the point. In any even dimension, such a manifold \(M\) with two fixed points exists, a rotation of an even dimensional sphere. Suppose that \(M\) has three fixed points. Then the dimension of \(M\) is a multiple of 4. Under the assumption that each isotropy submanifold is orientable, we show that if \(\dim M=8\), then the weights at the fixed points agree with those of an action on the quaternionic projective space \(\mathbb{HP}^2\), and show that there is no such 12-dimensional manifold \(M\).