Evaluate $\int_{0}^{\infty} t^{-1 / 2} e^{-1985\left(t+t^{-1}\right)} d t$. You may assume that $\int_{-\infty}^{\infty} e^{-x^{2}} d x=\sqrt{\pi}$. The original answer is in the form $\frac{\sqrt{\pi}}{\sqrt{m}} e^{-2 \times m}$ . Please provide the value of $m + 1$.

Let's think step by step and output the final answer within \boxed{}.
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