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Consider the following equations $\begin{aligned} &a_{1}+a_{2}+a_{3} &=& 6 \\ &a_{1}^2+a_{2}^2+a_{3}^2&=&194 \\ &a_{1}^3+a_{2}^3+a_{3}^3&=&126 \\ &a_{1}^4+a_{2}^4+a_{3}^4&=&13058 \\ \end{aligned}$

Find $a_{1}^5+a_{2}^5+a_{3}^5$.

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