
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 triangular pyramid \( A B C D \) is an equilateral triangle \( A B C \). The volume of the pyramid is \( \frac{100}{3 \sqrt{3}} \), and its height drawn from vertex \( D \) is 4. Point \( M \) is the midpoint of edge \( C D \). It is known that the radii of the spheres inscribed in pyramids \( A B C M \) and \( A B D M \) are equal to each other.

a) Find all possible values of the angle between the faces of the pyramid along edge \( A B \).

b) Find all possible values of the length of edge \( C D \), given additionally that faces \( B C D \) and \( A B C \) are mutually perpendicular.

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