Let \( T_1 \) and \( T_2 \) be two distinct common tangents to the ellipse \( E: \, \frac{x^2}{6} + \frac{y^2}{3} = 1 \) and the parabola \( P: \, y^2 = 12x \). Suppose that the tangent \( T_1 \) touches \( P \) and \( E \) at the points \( A_1 \) and \( A_2 \), respectively and the tangent \( T_2 \) touches \( P \) and \( E \) at the points \( A_4 \) and \( A_3 \), respectively. Then which of the following statements is(are) true?
