Minimum Distance on Reciprocal Curve

In a plane rectangular coordinate system $xOy$, the graph of function $y = \frac{1}{|x|}$ is $\Gamma$. Let points $P, Q$ on $\Gamma$ satisfy: $P$ is in the first quadrant, $Q$ is in the second quadrant, and line $PQ$ is tangent to the part of $\Gamma$ in the second quadrant at point $Q$. Find the minimum of $|PQ|$.