Nested Square Roots

Algebra Level 5

If \[\sqrt{4+\sqrt{4-\sqrt{4+\sqrt{4-\cdots}}}}=\frac{a+\sqrt{b}}{c}\] such that the fraction in the RHS cannot be simplified any further and \(b\) is square free and \(a,b,c\) are all integers, what is the value of \(a + b + c\)?

Extra Credit: Post a solution that can do it by hand without the requirement of quartic formula.

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