
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
Circles \(\omega_{1}, \omega_{2},\) and \(\omega_{3}\) are centered at \(M, N,\) and \(O\), respectively. The points of tangency between \(\omega_{2}\) and \(\omega_{3}\), \(\omega_{3}\) and \(\omega_{1}\), and \(\omega_{1}\) and \(\omega_{2}\) are \(A, B,\) and \(C\), respectively. Line \(MO\) intersects \(\omega_{3}\) and \(\omega_{1}\) again at \(P\) and \(Q\), respectively, and line \(AP\) intersects \(\omega_{2}\) again at \(R\). Given that \(ABC\) is an equilateral triangle of side length 1, compute the area of \(PQR\).

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