Given $x, y, z \in \mathbb{R}^+$, which are solutions to the system of equations:
\[
x^2 + xy + y^2 = 57
\]
\[
y^2 + yz + z^2 = 84
\]
\[
z^2 + zx + x^2 = 111
\]
What is the value of $xy + 3yz + 5zx$?

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