What Happened to Christmas?

Algebra Level 3

If

\[\sqrt{12 + 25\sqrt{12 + 25\sqrt{12 + 25\cdots}}} = \frac{a + \sqrt{b}}{c},\]

for positive integers \(a, b,\) and \(c\) such that \(b\) is not divisible by the square of any prime, find \(a+b+c.\)

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