Please reason step by step, and put your final answer within \boxed{}.
The graph of the function $f(x)=e^{x}+x^{2}+x+1$ and the graph of $g(x)$ are symmetric about the line $2x-y-3=0$. Let $P$ and $Q$ be the moving points on the graphs of $f(x)$ and $g(x)$, respectively. The minimum value of $|PQ|$ is \_\_\_\_\_\_\_\_.
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