Let \( \hat{a},\ \hat{b} \) be unit vectors. If \( \vec{c} \) be a vector such that the angle between \( \hat{a} \) and \( \vec{c} \) is \( \frac{\pi}{12} \), and \( \hat{b} = \vec{c} + 2(\vec{c} \times \hat{a}) \), then \( \lvert 6\,\vec{c} \rvert^{2} \) is equal to:
