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Algebra Level 4

Consider the following equations a1+a2+a3=6a12+a22+a32=194a13+a23+a33=126a14+a24+a34=13058\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 a15+a25+a35a_{1}^5+a_{2}^5+a_{3}^5.

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