I'll give you a hint, use the conversation of linear momentum equation.
For a fixed control volume $CV$:
$$ \sum \vec{F} = \frac{d}{dt} \int_{CV} \rho \vec{V} dV + \int_{CS} \rho \vec{V}(\vec{V}_r . \vec{n}) dA$$
wher $\vec{V}_r = \vec{V} - \vec{V}_{CS}$ is the relative flow velocity exiting the control volume relative to the control surface.
Assuming a steady state case:
$$ \sum \vec{F} =\int_{CS} \rho \vec{V}(\vec{V}_r . \vec{n}) dA = \dot{m} V_r$$
NOTE: your idea of using Bernoulli is justified if there was a difference in pressure between inlets and outlets of your control volume, and you needed to plug forces resulting from difference in pressure in the above Newton second law formulation, but since pressure at inlet = pressure at outlet = $p_{\infty}$, there is no need to use Bernoulli.