What is the asymptotic behaviour of $\max_{n \in \{1, 2,\, \dots N\}} \tan n$?

Question

What is the asymptotic behaviour of $f(N) = \max_{n \in \{1, 2,\, \dots N\}} \tan n$? Any non-trivial bounds above or below would be of interest. Some quick numerical experimentation shows:

>>> for N in [10, 100, 1000, 10000, 100000, 1000000, 10000000]:
...     f = max(map(math.tan, range(N)))
...     print "N = %8i, f(N) = %11.3f, f(N)/N = %r" % (N, f, f/N)
... 
N =       10, f(N) =       1.557, f(N)/N = 0.15574077246549023
N =      100, f(N) =       9.004, f(N)/N = 0.0900365494560708
N =     1000, f(N) =     229.071, f(N)/N = 0.22907138224446938
N =    10000, f(N) =     279.198, f(N)/N = 0.027919813178792136
N =   100000, f(N) =   40589.614, f(N)/N = 0.4058961353116651
N =  1000000, f(N) =  383610.708, f(N)/N = 0.3836107077437273
N = 10000000, f(N) = 3910993.434, f(N)/N = 0.391099343356971

The values sure look proportional to $N$, with a proportionality constant of maybe about $0.4$. To make my question explicit, I have two related queries:

1. If we simplify and model the first $N$ naturals as simply being $N$ i.i.d. uniform samples on $[0, 2\pi)$, then I imagine there is some nice way of interpreting $f(N)$ as an order statistic on i.i.d. draws from a Cauchy distribution, and maybe this yields the above proportionality exactly?

2. Can we place non-trivial bounds on $f(N)$ without resorting to such simplifications?

Edit: I can't reply to comments without 50 rep, but a comment points out that $f(N)$ is pretty clearly unbounded. When I say "asymptotic behavior" I'm not just asking for $\lim_{N \to \infty} f(N)$. I'm asking for a more subtle asymptotic analysis. Interesting answers would be things like "it follows from known results that $f(N)$ crosses $c N$ infinitely many times, for any $c$ in this range", "$f(N) \in \Omega(N)$", or "$f(N) \in O(\exp(N))$". Sorry, I hope this clarifies my question.