= Solution
Seek a constant transverse <guiding-centre drift> $\mathbf v_d$ whose magnetic <Lorentz force> balances the applied force. Define
$$
\mathbf v_d=\frac{\mathbf F\times\mathbf B}{qB^2}.
$$
Since $\mathbf F\cdot\mathbf B=0$, the vector triple-product identity gives $q\mathbf v_d\times\mathbf B=-\mathbf F$. Writing $\mathbf v=\mathbf v_d+\mathbf u$ therefore reduces the full equation to $m\dot{\mathbf u}=q\mathbf u\times\mathbf B$, the ordinary gyromotion equation. Its transverse orbit is circular in the drifting frame, so
$$
\boxed{\mathbf v_{\rm gc}=\mathbf v_d=\frac1q\frac{\mathbf F\times\mathbf B}{B^2}.}
$$
Averaging over a gyroperiod removes the transverse oscillatory velocity and leaves this drift, together with any constant parallel velocity. For an electric force $\mathbf F=q\mathbf E$, this becomes the charge-independent $\mathbf E\times\mathbf B/B^2$ drift; a charge-independent force instead gives opposite drift directions for opposite charge signs.
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