
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
In the angle of $60^\circ$ $X O Y$, a point $A$ is taken on the $O X$ arm such that $O A = a$. At point $A$, we draw a perpendicular to $O X$, which intersects $O Y$ at point $D$. Above $A D$, we construct a semicircle, which intersects $O Y$ at point $B$. Questions:

1) What is the length of $B D$?
2) What is the surface area of the solid generated by rotating $B D$ around $O X$?
3) What is the volume of the solid generated by rotating the triangle $A B D$ around $O X$?

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