Let \( \theta_{k}(k=1,2,3, \ldots \ldots, n), \theta_{1}\theta_{2}\...

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Let \( \theta_{k}(k=1,2,3, \ldots \ldots, n), \theta_{1}\theta_{2}\ldots . .\theta_{n} \) be the solution of \( \theta \) such that \( \sqrt{3} \sin n \theta+ \)
\( \mathrm{P} \) \( \cos n \theta=0 \). If \( \lim _{n \rightarrow \infty} \frac{1}{n} \sum_{k=1}^{n} \cos \frac{\theta_{k}}{2}=\frac{k}{\pi} \), then find the value of \( k(k \in N) \).

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