
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 a right prism is a regular hexagon with vertices $A_1, \ldots, A_6$ and the top base has vertices $B_1, \ldots, B_6$ with center $O$. The side edge $A_{1} B_{1}$ is equal to the diagonal $A_{1} A_{3}$. On the side edges $A_{1} B_{1}$, $A_{3} B_{3}$, and $A_{5} B_{5}$, consider points $C_{1}, C_{3}$, and $C_{5}$ respectively such that $A_{1} C_{1} = A_{3} C_{3} = A_{5} C_{5} = A_{1} A_{2}$. Determine the surface area and the volume of the prism's part below the planes $C_{1} O C_{3}$, $C_{3} O C_{5}$, and $C_{5} O C_{1}$ if $A_{1} A_{2} = 1$.

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