Expanding A Tetration Expression!

Algebra Level 5

\[ \large \prod_{k=1}^{2016} \left (1 + z_k x^{11^k} \right) = 1 + a_1 x^{m_1} + a_2 x^{m_2} + \cdots + a_{2017} x^{m_{2017}} + \cdots + a_N x^{m_N} \]

Consider the expansion above for \(z_k = e^{2k\pi i /11} \), where \(a_1 <a_2 < a_3 < \cdots < a_N\) and \(m_1 < m_2 < \cdots < m_N \) are all constants.

If \(z_n =a_{2017} \), find the minimum value of \(n\).

\[\]Clarification: \(i=\sqrt{-1}\).

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