# Calculating the module of a vector

I was doing a linear algebra assignment on vectors but I came across with some doubts. One of the problems says Find the module of a vector $$\vec{w}$$ if $$\vec{c} =(-2;3)$$ is perpendicular to $$\vec{a}$$ , $$\langle \vec{a} +2 \vec{w}$$ , $$\vec{c}\rangle =2 \sqrt{13}$$ and the angle formed by $$\vec{w}$$ and $$\vec{c}$$ is $$60^\circ$$

I don't know how to do it because it seems like there are many unknowns. I know that the product between two perpendicular vectors is equal to $$0$$. But how can I solve it?

Thanks