
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
Given a tetrahedron $ABCD$ with each vertex on the surface of sphere $O$, $AB=AC=5$, $BC=8$, $AD\perp$ the base plane $ABC$, $G$ is the centroid of $\triangle ABC$, and the tangent of the angle formed by line $DG$ and the base plane $ABC$ is $\dfrac{1}{2}$, then the surface area of sphere $O$ is \_\_\_\_\_\_.

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