
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
Let \(A B C D E\) be a square pyramid of height \(\frac{1}{2}\) with a square base \(A B C D\) of side length \(A B = 12\) (so \(E\) is the vertex of the pyramid, and the foot of the altitude from \(E\) to \(A B C D\) is the center of square \(A B C D\)). The faces \(A D E\) and \(C D E\) meet at an acute angle of measure \(\alpha\) (so that \(0^{\circ}<\alpha<90^{\circ}\)). Find \(\tan \alpha\).

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