An infinite geometric sequence $a_1,a_2,a_3,\dots$ satisfies $a_1=1$ and \[\frac{1}{a_1a_2}+\frac{1}{a_2a_3}+\frac{1}{a_3a_4}\cdots=\frac{1}{2}.\] The sum of all possible values of $a_2$ can be expressed as $m+\sqrt{n}$, where $m$ and $n$ are integers and $n$ is not a positive perfect square. Find $100m+n$.

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