
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 circle with radius equal to the length of the legs of a right isosceles triangle $ABC$ (where $\angle ABC = 90^\circ$) is placed such that it rotates about vertex $B$ and is tangent to side $BC$. As the circle rotates, it intersects line $AC$ at a variable point $M$. Let $m$ be the number of degrees in arc $MBM$ where $M'$ is the reflection of point $M$ across line $BC$. Calculate the possible degrees for arc $MBM'$ as the circle makes a complete rotation about $B$.
A) $90^\circ$  
B) $120^\circ$  
C) $180^\circ$  
D) Varies depending on the angle of rotation  
E) None of these

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