\[\LARGE x^{x^{x^{\cdot^{\cdot^\cdot}}}} = 1, \qquad x^{x^{x^{\cdot^{\cdot^\cdot}}}} = 2,\qquad x^{x^{x^{\cdot^{\cdot^\cdot}}}} = 3, \qquad x^{x^{x^{\cdot^{\cdot^\cdot}}}} = 4 \]

The above shows 4 equations. How many of these infinitely nested equations yield a solution for \(x?\)

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