Quotient of $S^3\times S^3$ by an action of $S^1$

Consider the action of $S^1$ on the product of 3-spheres $S^3\times S^3$ defined by:

$$e^{it}.(z_1, z_2)=(e^{2it}z_1, e^{3it}z_2)$$

where $z_1, z_2\in S^3$. Here we understant $e^{2it}z_1$ as the multiplication by $e^{2it}$ in each component of $z_1$ when we look at $S^3$ as a subspace of $\mathbb{C}^2$ (more precisely, $S^3=\{(a,b)\in \mathbb{C}^2: |a|^2+|b|^2=1\}$).

So my question is what is (topologically, for instance) the quotient of $S^3\times S^3$ by this action? I'm convinced that this should give $S^3\times S^2$ because this action is in some sense a "twisted" Hopf fibration, but I've not been able to show this (at least in an explicit way).

Ivy's answer is completely correct, but one would have to show that $S^3\times S^3/S^1$ is spin for this particular action of $S^1$ on $S^3\times S^3$. It is, and, in fact, for any homogeneous action of of $S^1$ on $S^3\times S^3$, the quotient is diffeomorphic to $S^2\times S^3$. (Here, homogeneous means the action can be written as $z(p,q) = (z^a p, z^b q)$ for $p,q\in \mathbb{H}$, $a,b\in \mathbb{Z}$, $\operatorname{gcd}(a,b) = 1$).

As Ivy says, this follows from the Barden-Smale classification of simply connected $5$-manifolds. In fact, for most choices of $(a,b)$, no one knows how to write down an explicit diffeomorphism. (The only exceptions I know are $(a,b)\in \{\pm (1,0), \pm (0,1), (\pm 1,\pm 1)\}$). My advisor, Wolfgang Ziller, has offered to pay for dinner at any restaurant in Philadelphia for anyone who figures out such a diffeomorphism for general $(a,b)$.

Such a diffeomorphism must use the fact that the action is homogeneous, because (as Mike mentioned in the comments) there are free linear actions of $S^1$ on $S^3\times S^3$ whose quotients are not diffeomorphic to $S^2\times S^3$. That said, only one other diffeomorphism type can arise: the unique non-trivial linear $S^3$ bundle over $S^2$.

In case you care, here is a summary of the results:

A general linear action of $S^1$ on $S^3\times S^3$ is equivalent to one of the form $z\ast(p,q) = (z^a p z^c, z^b q z^d)$ for integers $(a,b,c,d)\in \mathbb{Z}^4$. If $\operatorname{gcd}(a,b,c,d) \neq 1$, we have an ineffective kernel we can divide out, so we assume $\operatorname{gcd}(a,b,c,d) = 1$. The action is effectively free iff $\operatorname{gcd}(a^2 - c^2, b^2 - d^2) \in \{1,2\}$. If this $\operatorname{gcd}$ is $1$, the quotient is $S^2\times S^3$ and other wise, the quotient is non-trivial $S^3$ bundle over $S^2$.

And I apologize for the self promotion, but all of this was worked on in my thesis - the relevant paper can be found at https://arxiv.org/abs/1304.1770.

Actually, for a general action $e^{it}.(z_1,z_2)=(e^{ikt}z_1,e^{ilt}z_2)$ where $(k,l)=1$, the quotient is $S^2\times S^3$. Just use Smale's classification results of simply connected spin $5$-manifolds. For more details, see for example

Einstein metrics on principal torus bundles, by M. Wang and W. Ziller.