I heard you liked infinite series, so here's an infinite series in an infinite series!

Calculus Level 4

\[ 1 = \dfrac{(1-a) - \dfrac{(1-a)^2}{2} + \dfrac{(1-a)^3}{3} - \dfrac{(1-a)^4}{4} + ...}{(1-b) - \dfrac{(1-b)^2}{2} + \dfrac{(1-b)^3}{3} - \dfrac{(1-b)^4}{4} + ... } \]

where \(\begin{cases} a = 1 - \dfrac{(15x)^2}{2!} + \dfrac{(15x)^4}{4!} - \dfrac{(15x)^6}{6!} + ... \\ b = 15x - \dfrac{(15x)^3}{3!} + \dfrac{(15x)^5}{5!} - \dfrac{(15x)^7}{7!} +... \end{cases} \) for \(x \in [0, 2\pi ] \).

How many real solutions for \( x \) exist to make the top-most equation true?

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