
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 non-equilateral triangle  $\triangle ABC$   of perimeter  $12$  is inscribed in circle  $\omega$  .Points  $P$  and  $Q$  are arc midpoints of arcs  $ABC$  and  $ACB$  , respectively. Tangent to  $\omega$  at  $A$  intersects line  $PQ$  at  $R$ .
It turns out that the midpoint of segment  $AR$  lies on line  $BC$  . Find the length of the segment  $BC$ .

*(А. Кузнецов)*

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