Sum and Square Sum

Algebra Level 5

\( (x, y, z) \in \mathbb{R}^3\) are points that lie on the plane \( x + 2y + 3z = 78\), and lie on the sphere \(x^2 + y^2 + z^2 = 468\). The maximum value of \(x\) has the form \( \frac {a}{b} \), where \(a\) and \(b\) are coprime positive integers. What is the value of \( a+b\)?

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