Prove that the limit of the ratio of maximum to minimum summed projections of infinite vectors is 1?

Question

Let $X$ be a set of $n$ randomly oriented vectors in $\mathbf{R}^2$. Define the function $f$ as the ratio \begin{equation} f=\frac{\min_{\theta'}\sum_{n}|\| \vec{x_n} \|\cos{(\theta_{n}-\theta')}|}{\max_{\theta'}\sum_{n}|\| \vec{x_n} \|\cos{(\theta_{n}-\theta')}|} \end{equation} where the $\theta_n$ denote the orientation of each of the vectors $\vec{x_n}$, and the $\theta'$ are the angles in $[0,\pi]$. In other words, $f$ represents the ratio of the sum of the projection of the $n$ vectors at the angle yielding the minimal sum over the sum at the angle yielding the maximal sum. Intuitively, I think the expected value of $f$ ought to approach 1 as $n$, the number of vectors approaches infinity, but I am having trouble showing this. I would ideally like to find a closed form solution of the expected value of $f$ as a function of $n$. When I plot out simulated values of $f$ vs $n$ it looks like a nice analytical function with that limit. plot of simulation

Here is what I have attempted: I am not sure this is the right way to go about this problem.
Define $U=\| \vec{x_n} \|\sim\mathrm{Lognormal}(0,\sigma^2)$ (assume that the lengths of the vectors are lognormally distributed) and $V=\cos{(\theta_{n}-\theta)}$. Now cosine is a non-monotonic function with two inverse functions $\arccos(\theta)$ and $-\arccos(\theta)$. By applying the change of variables formula for $V$ and assuming that the angles $\theta_n$ of each of the vectors is uniformly distributed in $[0,\pi]$ (that is, $f_{\theta_n}\sim\frac{1}{\pi}$, I get that the p.d.f. of $V$ is \begin{equation}\begin{split} f_V(V)&= f_{\theta_n}(\theta'+\arccos(V))|\frac{d}{dV}(\theta'+\arccos(V))| + f_{\theta_n}(\theta'-\arccos(V))|\frac{d}{dV}(\theta'-\arccos(V))|\\ &=\frac{f_{\theta_n}(\theta'+\arccos(V)) + f_{\theta_n}(\theta'-\arccos(V))}{\sqrt{1-V^2}} \end{split} \end{equation} Since we are assuming that the vectors' lengths and orientations are independently distributed, the joint distribution would be the product of the lognormal and $f_V(V)$, and to get the distribution of the sum of $n$ terms, we would take the product of that distribution $n$ times. But then I get stuck. I'm not sure how to get the expected value of the minimum or maximum as these aren't simple extrema of a set of discrete values. Am I going about this problem in completely the wrong way, or is there a way to solve the limit using this method?