
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 $K L M N$ with a volume of 75 is the equilateral triangle $K L M$ with a side length of 10. Point $T$ is the midpoint of edge $M N$. It is known that the radii of the spheres inscribed in pyramids $K L M T$ and $K L N T$ are equal.

a) Find all possible values of the angle between the faces of the pyramid at edge $K L$.

b) Find all possible values of the length of edge $M N$, if it is additionally known that faces $K L M$ and $L M N$ are perpendicular.

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