Let $$\theta$$ be the angle between the vectors $$\vec{a}$$ and $$\vec{b}$$, where $$|\vec{a}| = 4, |\vec{b}| = 3$$ and $$\theta \in \left[\frac{\pi}{4}, \frac{\pi}{3}\right]$$. Then $$|\vec{a} - \vec{b} \times \vec{a} + \vec{b}|^2 + 4|\vec{a} \cdot \vec{b}|^2$$ is equal to ______.

Given $$|\vec{a}| = 4$$, $$|\vec{b}| = 3$$ and $$\theta \in \left[\frac{\pi}{4}, \frac{\pi}{3}\right]$$, we wish to determine the value of $$|(\vec{a} - \vec{b}) \times (\vec{a} + \vec{b})|^2 + 4|\vec{a} \cdot \vec{b}|^2$$.

Expanding the cross product, we have $$(\vec{a} - \vec{b}) \times (\vec{a} + \vec{b}) = \vec{a} \times \vec{a} + \vec{a} \times \vec{b} - \vec{b} \times \vec{a} - \vec{b} \times \vec{b}.$$ Since $$\vec{a} \times \vec{a} = \vec{0}$$, $$\vec{b} \times \vec{b} = \vec{0}$$, and $$\vec{b} \times \vec{a} = -\,\vec{a} \times \vec{b},$$ this reduces to $$2(\vec{a} \times \vec{b}).$$

Therefore, the squared magnitude of this cross product is $$|2(\vec{a} \times \vec{b})|^2 = 4\,|\vec{a} \times \vec{b}|^2 = 4\,|\vec{a}|^2\,|\vec{b}|^2\sin^2\theta.$$

Similarly, the second term is $$4\,|\vec{a} \cdot \vec{b}|^2 = 4\,|\vec{a}|^2\,|\vec{b}|^2\cos^2\theta.$$

Adding these results gives $$4\,|\vec{a}|^2\,|\vec{b}|^2\bigl(\sin^2\theta + \cos^2\theta\bigr) = 4\,|\vec{a}|^2\,|\vec{b}|^2 = 4 \times 16 \times 9 = 576.$$ Hence, the required value is $$576$$.