
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 lower base of the prism $A B C A_1 B_1 C_1$ with side edges $A A_1$, $B B_1$, $C C_1$ is an equilateral triangle $A B C$ with a side length of $2a$. The projection of the prism onto the plane of the base is a trapezoid with a lateral side $A B$ and an area twice that of the base. The radius of the sphere passing through the vertices $A$, $B$, $A_1$, $C_1$ is $2a$. Find the volume of the prism. (Find all solutions).

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