
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 cube ABCD-A<sub>1</sub>B<sub>1</sub>C<sub>1</sub>D<sub>1</sub>, E is a point on A<sub>1</sub>B<sub>1</sub>. If the tangent of the acute dihedral angle formed by plane EBD and plane ABCD is $\frac {3 \sqrt {2}}{2}$, and if the diameter of the circumscribed sphere of the tetrahedron A-A<sub>1</sub>D<sub>1</sub>E is denoted as $a$, then $\frac {a}{|AB|}=$ \_\_\_\_\_\_.

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