
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
Each of three circles $\omega_1$, $\omega_2$, and $\omega_3$ has radius 3, and are arranged in the plane so that each circle is externally tangent to the other two. Define points $P_1$ on $\omega_1$, $P_2$ on $\omega_2$, and $P_3$ on $\omega_3$ such that $P_1P_2=P_2P_3=P_3P_1$ and line $P_iP_{i+1}$ is tangent to circle $\omega_i$ for each $i=1,2,3$ with $P_4 = P_1$. Find the area of $\triangle P_1P_2P_3$ expressed in the form $\sqrt{a}+\sqrt{b}$ for positive integers $a$ and $b$, and determine the sum $a+b$.

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