Fractal from Sqrt!

Algebra Level 3

\[\Large a + \dfrac b{c + \dfrac d{a + \dfrac b{c+\dfrac d{a + \dfrac b{\ddots}}}}} = \sqrt 6 \]

The equation above holds true for some positive integers \(a\), \(b=1\), \(c\), and \(d\). Find \( a + b + c + d \).

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