<|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
 In a quadrilateral pyramid \( S A B C D \), the base \( A B C D \) has its axis of symmetry along the diagonal \( A C \), which is 9 units long. The point \( E \), where the diagonals of quadrilateral \( A B C D \) intersect, divides the segment \( A C \) such that \( A E \) is shorter than \( E C \). A plane is passed through the midpoint of a lateral edge of the pyramid \( S A B C D \), parallel to the base, intersecting the sides \( S A, S B, S C, \) and \( S D \) at points \( A_1, B_1, C_1, \) and \( D_1 \) respectively.

The resulting polyhedron \( A B C D A_1 B_1 C_1 D_1 \), which is part of the pyramid \( S A B C D \), intersects the plane \(\alpha\) forming a regular hexagon with side length 2. Find the area of triangle \(\triangle A B D\) if plane \(\alpha\) intersects the segments \( B B_1 \) and \( D D_1 \). 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
4
In a quadrilateral pyramid \( S A B C D \), the base \( A B C D \) has its axis of symmetry along the diagonal \( A C \), which is 9 units long. The point \( E \), where the diagonals of quadrilateral \( A B C D \) intersect, divides the segment \( A C \) such that \( A E \) is shorter than \( E C \). A plane is passed through the midpoint of a lateral edge of the pyramid \( S A B C D \), parallel to the base, intersecting the sides \( S A, S B, S C, \) and \( S D \) at points \( A_1, B_1, C_1, \) and \( D_1 \) respectively.

The resulting polyhedron \( A B C D A_1 B_1 C_1 D_1 \), which is part of the pyramid \( S A B C D \), intersects the plane \(\alpha\) forming a regular hexagon with side length 2. Find the area of triangle \(\triangle A B D\) if plane \(\alpha\) intersects the segments \( B B_1 \) and \( D D_1 \).
test