
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
A cube with an edge of length \( a = \frac{1}{\sqrt{2}} \) is illuminated by a cylindrical beam of light with a radius \( \rho = \sqrt{2 - \sqrt{3}} \). The beam is directed along the main diagonal of the cube (the axis of the beam contains the main diagonal). Find the area of the illuminated part of the cube's surface.

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