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Let

\{\varepsilon_i\}_{i\ge 1}, \{\theta_i\}_{i\ge 0}

be two infinite sequences of real numbers, such that

\varepsilon_i \in \{-1,1\}

for all

i

, and the numbers

\theta_i

obey

\tan \theta_{n+1} = \tan \theta_{n}+\varepsilon_n \sec(\theta_{n})-\tan \theta_{n-1} , \qquad n \ge 1

and

\theta_0 = \frac{\pi}{4}, \theta_1 = \frac{2\pi}{3}

. Compute the sum of all possible values of

\lim_{m \to \infty} \left(\sum_{n=1}^m \frac{1}{\tan \theta_{n+1} + \tan \theta_{n-1}} + \tan \theta_m - \tan \theta_{m+1}\right)

Proposed by Grant Yu

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