GAIC Math 2024 Problem 84

Given a hyperbola $\Gamma: \frac{x^{2}}{7}-\frac{y^{2}}{5}=1$, a line $l: a x+b y+1=0$ intersects $\Gamma$ at point $A$. A tangent to $\Gamma$ drawn through point $A$ is perpendicular to the line $l$. Then $\frac{7}{a^{2}}-\frac{5}{b^{2}}=$_____.