For some positive integer $n,$ $0 < n < 180,$
\[\csc (2^3)^\circ + \csc (2^4)^\circ + \csc (2^5)^\circ + \dots + \csc (2^{2019})^\circ = \sec n^\circ.\]Find $n.$
Question:
For some positive integer $n,$ $0 < n < 180,$
\[\csc (2^3)^\circ + \csc (2^4)^\circ + \csc (2^5)^\circ + \dots + \csc (2^{2019})^\circ = \sec n^\circ.\]Find $n.$
Answer:
Let's think step by step.

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For some positive integer $n,$ $0 < n < 180,$
\[\csc (2^3)^\circ + \csc (2^4)^\circ + \csc (2^5)^\circ + \dots + \csc (2^{2019})^\circ = \sec n^\circ.\]Find $n.$
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