
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 tangent at  $A$ ,  $B$ , 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}
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