
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
The base of the triangular pyramid \(ABCD\) is the triangle \(ABC\) in which \(\angle A = \frac{\pi}{2}\), \(\angle C = \frac{\pi}{6}\), and \(|BC| = 2\sqrt{2}\). The lengths of the edges \(AD\), \(BD\), and \(CD\) are equal. A sphere with radius 1 touches the edges \(AD\), \(BD\), the extension of the edge \(CD\) beyond point \(D\), and the plane \(ABC\). Find the length of the tangent segment drawn from point \(A\) to the sphere.

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