<|im_start|>system
You are a helpful assistant. You first think about the reasoning process in the mind and then provide the user with the answer. You should try to truely solve the question by yourself without cheating.  <|im_end|>
<|im_start|>user
 As shown in the figure, on the line \( m \), three equilateral triangles are placed: \(\triangle ABC\), \(\triangle HFG\), and \(\triangle DCE\). It is known that \(BC = \frac{1}{3} CE\), and \(F\) and \(G\) are the midpoints of \(BC\) and \(CE\) respectively. \(FM \parallel AC\) and \(GN \parallel DC\). Let the areas of the three parallelograms in the figure be \(S_{1}, S_{2}, S_{3}\), respectively. If \(S_{1} + S_{3} = 10\), then \(S_{2} =\) ___? Show your work in <think> </think> tags. And return the final answer in <answer> </answer> tags.<|im_end|>
<|im_start|>assistant
Let me solve this step by step.
<think>
user
3
As shown in the figure, on the line \( m \), three equilateral triangles are placed: \(\triangle ABC\), \(\triangle HFG\), and \(\triangle DCE\). It is known that \(BC = \frac{1}{3} CE\), and \(F\) and \(G\) are the midpoints of \(BC\) and \(CE\) respectively. \(FM \parallel AC\) and \(GN \parallel DC\). Let the areas of the three parallelograms in the figure be \(S_{1}, S_{2}, S_{3}\), respectively. If \(S_{1} + S_{3} = 10\), then \(S_{2} =\) ___?
test