
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 point  $O$  is choosen inside a triangle  $ABC$  so that the length of segments  $OA$ ,  $OB$  and  $OC$  are equal to  $15$ , $12$  and  $20$ , respectively. It is known that the feet of the perpendiculars from  $O$  to the sides of the triangle  $ABC$  are the vertices of an equilateral triangle.
 Find the value of the angle  $BAC$ .

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