Find the positive integer $N$ for which there exist reals $\alpha, \beta, \gamma, \theta$ which obey the following equations:
\[
\begin{align*}
0.1 &= \sin \gamma \cos \theta \sin \alpha, \\
0.2 &= \sin \gamma \sin \theta \cos \alpha, \\
0.3 &= \cos \gamma \cos \theta \sin \beta, \\
0.4 &= \cos \gamma \sin \theta \cos \beta, \\
0.5 &\ge \left\lvert N-100 \cos 2\theta \right\rvert.
\end{align*}
\]

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