Find the minimum value of
\[2 \cos \theta + \frac{1}{\sin \theta} + \sqrt{2} \tan \theta\]for $0 < \theta < \frac{\pi}{2}.$The answer is in the form k\sqrt{m}+n,. Please provide the value of k + m + n.

Let's think step by step and output the final answer within \boxed{}.
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