
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
Consider a circle  $k$  with center  $S$  and radius  $r$ . Denote  $\mathsf M$  the set of all triangles with incircle  $k$  such that the largest inner angle is twice bigger than the smallest one. For a triangle  $\mathcal T\in\mathsf M$  denote its vertices  $A,B,C$  in way that  $SA\ge SB\ge SC$ . Find the locus of points  $\{B\mid\mathcal T\in\mathsf M\}$ .

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