On the coordinate plane, let $L _ { 1 }$ and $L _ { 2 }$ be two lines passing through point $(3, 1)$ with slopes $m$ and $- m$ respectively, where $m$ is a real number. Let $\Gamma$ be a circle with center at the origin. Given that $\Gamma$ intersects $L _ { 1 }$ at two distinct points $A$ and $B$, and the distance from the center to $L _ { 1 }$ is 1, and $\Gamma$ is tangent to $L _ { 2 }$, then the length of chord $\overline { A B }$ is (express as a fraction in lowest terms).
On the coordinate plane, let $L _ { 1 }$ and $L _ { 2 }$ be two lines passing through point $(3, 1)$ with slopes $m$ and $- m$ respectively, where $m$ is a real number. Let $\Gamma$ be a circle with center at the origin. Given that $\Gamma$ intersects $L _ { 1 }$ at two distinct points $A$ and $B$, and the distance from the center to $L _ { 1 }$ is 1, and $\Gamma$ is tangent to $L _ { 2 }$, then the length of chord $\overline { A B }$ is \underline{\hspace{2cm}} (express as a fraction in lowest terms).