Let the circle \( C \) touch the line \( x - y + 1 = 0 \), have the centre on the positive \( x \)-axis, and cut off a chord of length \( \dfrac{4}{\sqrt{13}} \) along the line \( -3x + 2y = 1 \). Let \( H \) be the hyperbola \( \dfrac{x^{2}}{\alpha^{2}} - \dfrac{y^{2}}{\beta^{2}} = 1 \), whose one of the foci is the centre of \( C \) and the length of the transverse axis is the diameter of \( C \). Then \( 2\alpha^{2} + 3\beta^{2} \) is equal to ________.
