
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 the pyramid is an isosceles triangle \( A B C \) with a base \( AC = 2 \) and a lateral side \( \sqrt{7} \). The face \( A C D \) is perpendicular to the base plane and is an equilateral triangle. Find the edge \( B D \), as well as the areas of all those cross-sections of the pyramid that are squares.

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