You are a helpful assistant that can solve the given question step by step with the help of the wikipedia search tool and python interpreter tool. Given a question, you need to first think about the reasoning process in the mind and then provide the answer. During thinking, you can invoke the wikipedia search tool to search and python interpreter tool to calculate the math problem for fact information about specific topics if needed. The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags respectively, and the search query and result are enclosed within <search> </search> and <result> </result> tags respectively. For example, <think> This is the reasoning process. </think> <search> search query here </search> <result> search result here </result> <think> This is the reasoning process. </think> <python> python code here </python> <result> python interpreter result here </result> <think> This is the reasoning process. </think> <answer> The final answer is \[ oxed{answer here} \] </answer>. In the last part of the answer, the final exact answer is enclosed within oxed{} with latex format.
system
The base of pyramid \( TABC \) is a triangle \( ABC \) with all sides equal to \(\sqrt{3}\), and the height of the pyramid coincides with the lateral edge \( TA \). Find the area of the cross-section of the pyramid by a plane that passes through the center of the circumscribed sphere around the pyramid, forms an angle of \(60^\circ\) with the base plane, intersects the edge \( AB \) at point \( M \) such that \( MB = 2AM \), and intersects the edge \( BC \). It is given that the distance from point \( A \) to the plane of the cross-section is 0.25.
user
\frac{11 \sqrt{3}}{30}
rule
The base of pyramid \( TABC \) is a triangle \( ABC \) with all sides equal to \(\sqrt{3}\), and the height of the pyramid coincides with the lateral edge \( TA \). Find the area of the cross-section of the pyramid by a plane that passes through the center of the circumscribed sphere around the pyramid, forms an angle of \(60^\circ\) with the base plane, intersects the edge \( AB \) at point \( M \) such that \( MB = 2AM \), and intersects the edge \( BC \). It is given that the distance from point \( A \) to the plane of the cross-section is 0.25.
train