A $39$-tuple of real numbers $(x_1,x_2,\ldots, x_{39})$ satisfies the following conditions:
\[2\sum_{i=1}^{39} \sin(x_i) = \sum_{i=1}^{39} \cos(x_i) = -34.\]
Determine the ratio between the maximum of $\cos(x_1)$ and the maximum of $\sin(x_1)$ over all tuples $(x_1,x_2,\ldots, x_{39})$ satisfying the given conditions. This ratio is expressed as $\frac{a}{b}$ for coprime positive integers $a$ and $b$. Find the sum $a + b$.

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