If \( \vec{a} \) and \( \vec{b} \) make angle \( \cos^{-1}\!\left(\dfrac{5}{9}\right) \) with each other, then \( \lvert \vec{a} + \vec{b} \rvert = \sqrt{2} \, \lvert \vec{a} - \vec{b} \rvert \) for \( \lvert \vec{a} \rvert = n \, \lvert \vec{b} \rvert \). The integer value of \( n \) is ________.
