Weird exponent

Level pending

The value of \(x\) that satisfies the equation

\(8^{x^{x^{x^\cdots}}}=4\)

can be expressed as \(\frac{a\sqrt{b}}{c}\) where \(a,b,\) and \(c\) are positive integers, \(a\) and \(c\) are relatively prime, and \(b\) is not the square of any prime. Find \(abc\).

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