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This will give you the 4a, as cos(omega.x) = 0 => omega.x = pi/2 +- n.pi

Aren't you assuming A=0 for that solution?
\[Asin(\pi/2 \pm n\pi) \ne 0\]

*\[\varphi'(a)=\varphi'(-a)=0\]

That guess came from avoiding making the first derivative jump about at +- a