homework and exercises - Prove a force driven by a cross product between a vector and its velocity gives a spiral movement parallel to the vector

I was given a problem in a Classical Mechanics course that went somehow like the following:

"Consider a particle of mass $m$ moving under the presence of a force $\vec{F} = k\hat{x}\times\vec{v}$, where $\hat{x}$ is an unit vector in the positive direction of the $x$ axis and $k$ any constant. Prove that the movement of a particle under this force is restricted to a circular motion with angular velocity $\vec{\omega}=(k/m)\hat{x}$ or, in a more general case, a spiral movement parallel to the direction of $\hat{x}$."

In an Electrodynamics elementary college course you can see and solve that a magnetic force sort of as:

$$m\ddot{\vec{r}}=\frac{q}{c}\left(\vec{v}\times\vec{B}\right)$$

with a magnetic field, say, $\vec{B}=B_0\hat{z}$ can drive a particle through a spiral movement in the precise direction of that magnetic field you customize, involving a cyclotron frequency and so, if and only if you input further initial conditions to the movement in x, y and z.

My inquiry then is, how can you prove the relation given above for the angular velocity and conclude a spiral movement, from a classical mechanics perspective? I can see there's a link between both procedures, but I cannot try solving the first one without giving a glimpse to the latter.