The polynomial $f(x) = x^{52} + a_1x^{51} + a_2x^{50} + \cdots + a_{51}x+a_{52}$ has roots $\tan\theta_1, \tan\theta_2, \tan\theta_3, \ldots, \tan\theta_{52}.$

We are also told that:

- $f(-i) = i(2 + \sqrt3) + 1$,
- $f(-1) = i\sqrt3 + 1$,
- $f(i) = i(2 + \sqrt3) - 1$,
- $f(1) = i\sqrt3 - 1$, and
- $f(0) = 1$.

Find the minimum positive value of $\theta_1 + \theta_2 +\theta_3 + \cdots +\theta_{52}.$