# Many Variables

Algebra Level 5

\begin{align*} x + y + z &= 1, \\ xa^2 + yb^2 + zc^2 &= w^2, \\ xa^3 + yb^3 + zc^3 &= w^3, \\ xa^4 + yb^4 + zc^4 &= w^4 \end{align*}

The real numbers $$w$$, $$a$$, $$b$$, $$c$$ are distinct, such that there exist real numbers $$x$$, $$y$$, and $$z$$ satisfying the system of equations above.

If $$(a,b,c) = (13, 17, 19)$$, there is a unique value of $$|w|$$ such that it is less than 6. Find the value to the nearest tenth.

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