
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) A point $M$ runs around the circumcircle of a triangle, and we consider its projections on the sides of the triangle at every position. We find that $8/10$ of the positions of $M$ on the circle correspond to cases where all three projections fall on the extensions of the sides (or exactly on a vertex). What is the largest angle of the triangle?

b) A point $N$ runs around the incircle of a triangle. We find that $8/10$ of the positions of $N$ on the circle correspond to cases where one of the three projections falls on the extension of a side. What is the largest angle of the triangle? (It is sufficient to give the angle to an accuracy of $0.1^{\circ}$.

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