
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 perpendicular bisector through point \( A \) in shape \( ABCDA^{\prime}B^{\prime}C^{\prime}D^{\prime} \) intersects the edges \( AD \), \( AB \), and \( AA \) at angles such that these edges are equal to the side lengths of an inscribed regular hexagon, square, and triangle with radius \( R \).

1. Calculate the volume \( V \) of the shape and the diagonal \( AC^{\prime} \).
2. Calculate the volume and surface area of the shape that is formed when the quadrilateral \( AA^{\prime}C^{\prime}C \) is rotated around a line passing through point \( A^{\prime} \) and parallel to the diagonal \( AC^{\prime} \).
3. Calculate the value of \( R \) if \( V = 0.034786 \) m\(^3\).

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