
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 that points $A$, $B$, $C$ are on the surface of sphere $O$, and $AB$=$AC$=$1$, $\angle BAC$=$120^{\circ}$, the projection of point $P$ on the surface of the sphere on plane $ABC$ is exactly the circumcenter ${O_{1}}$ of $\triangle ABC$, and the volume of the conical frustum $P-ABC$ is $\frac{\sqrt{3}}{6}$, find the surface area of sphere $O$.

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