Find the solutions to
\[\frac{1}{x - \tan 20^{\circ}} + \frac{1}{x + \tan 40^{\circ}} + \frac{1}{x - \tan 80^{\circ}} = 0.\]Enter the solutions, separated by commas.
<|im_start|>system
You are a helpful assistant.<|im_end|>
<|im_start|>user
Find the solutions to
\[\frac{1}{x - \tan 20^{\circ}} + \frac{1}{x + \tan 40^{\circ}} + \frac{1}{x - \tan 80^{\circ}} = 0.\]Enter the solutions, separated by commas.
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant

user
2 + \sqrt{3}, -2 + \sqrt{3}
Find the solutions to
\[\frac{1}{x - \tan 20^{\circ}} + \frac{1}{x + \tan 40^{\circ}} + \frac{1}{x - \tan 80^{\circ}} = 0.\]Enter the solutions, separated by commas.
train