
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
In the triangular pyramid \( A-BCD \), \(\triangle BCD\) and \(\triangle ACD\) are both equilateral triangles with side lengths of 2. \(\triangle BCD\) lies in plane \(\alpha\), and the side \( AB = \sqrt{3} \). Four of the pyramid's vertices are randomly labeled with the numbers 1 to 8. Let the corresponding labels be \( f(\eta) \), where \(\eta\) takes values from \( A, B, C, D \). Point \( E \) lies on the side \( AB \). If \( \frac{|BE|}{|EA|} = \frac{f(B)}{f(A)}\), what is the probability that the dihedral angle \( E-CD-A \) has a plane angle \(\theta\) greater than \(\frac{\pi}{4}\)?

Present the answer in LaTex format: \boxed{Your answer}
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