SMO 2023 Open Round 1 Question 19

Let $ABC$ be a triangle with $AB=c$, $AC=b$ and $BC=a$, and satisfies the conditions $\tan C=\dfrac{\sin A+\sin B}{\cos A+\cos B}$, $\sin(B-A)=\cos C$ and that the area of the triangle $ABC=3+\sqrt3$. Determine the value of $a^2+c^2$.