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

Calculus Level 4

1=(1a)(1a)22+(1a)33(1a)44+...(1b)(1b)22+(1b)33(1b)44+... 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 {a=1(15x)22!+(15x)44!(15x)66!+...b=15x(15x)33!+(15x)55!(15x)77!+...\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[0,2π]x \in [0, 2\pi ] .

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

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