Let \( \vec{c} \) and \( \vec{d} \) be vectors such that \( |\vec{c} + \vec{d}| = \sqrt{29} \) and \( \vec{c} \times (2\hat{i} + 3\hat{j} + 4\hat{k}) = (2\hat{i} + 3\hat{j} + 4\hat{k}) \times \vec{d} \). If \( \lambda_1, \lambda_2 \) \( (\lambda_1 > \lambda_2) \) are the possible values of \( (\vec{c} + \vec{d}) \cdot (-7\hat{i} + 2\hat{j} + 3\hat{k}) \), then the equation \( K^2 x^2 + (K^2 - 5K + \lambda_1)xy + \left(3K + \frac{\lambda_2}{2}\right) y^2 - 8x + 12y + \lambda_2 = 0 \) represents a circle, for \( K \) equal to:
