
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
An acute triangle  $ABC$  is inscribed in a circle of radius 1 with centre  $O;$  all the angles of  $ABC$  are greater than  $45^\circ.$   $B_{1}$  is the foot of perpendicular from  $B$  to  $CO,$   $B_{2}$  is the foot of perpendicular from  $B_{1}$  to  $AC.$  
Similarly,  $C_{1}$  is the foot of perpendicular from  $C$  to  $BO,$   $C_{2}$  is the foot of perpendicular from  $C_{1}$  to  $AB.$  
The lines  $B_{1}B_{2}$  and  $C_{1}C_{2}$  intersect at  $A_{3}.$  The points  $B_{3}$  and  $C_{3}$  are defined in the same way. 
Find the circumradius of triangle  $A_{3}B_{3}C_{3}.$ *Proposed by F.Bakharev, F.Petrov*

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