
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 \( DABC \) is the triangle \( ABC \), in which angle \( A \) is a right angle, angle \( ACB \) equals \( 30^{\circ} \), 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 edge \( CD \) beyond point \( D \), and the plane \( BAC \). Construct the image of the center \( O \) of this sphere and 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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