Let
\[a_n = \sqrt{1 + \left( 1 + \frac{1}{n} \right)^2} + \sqrt{1 + \left( 1 - \frac{1}{n} \right)^2}.\]Compute
\[\frac{1}{a_1} + \frac{1}{a_2} + \frac{1}{a_3} + \dots + \frac{1}{a_{100}}.\]
<|im_start|>system
You are a helpful assistant.<|im_end|>
<|im_start|>user
Let
\[a_n = \sqrt{1 + \left( 1 + \frac{1}{n} \right)^2} + \sqrt{1 + \left( 1 - \frac{1}{n} \right)^2}.\]Compute
\[\frac{1}{a_1} + \frac{1}{a_2} + \frac{1}{a_3} + \dots + \frac{1}{a_{100}}.\]
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant

user
\frac{\sqrt{20201} - 1}{4}
Let
\[a_n = \sqrt{1 + \left( 1 + \frac{1}{n} \right)^2} + \sqrt{1 + \left( 1 - \frac{1}{n} \right)^2}.\]Compute
\[\frac{1}{a_1} + \frac{1}{a_2} + \frac{1}{a_3} + \dots + \frac{1}{a_{100}}.\]
train