
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 $\omega_3$ each have radius $5$ and are positioned such that each circle is externally tangent to the other two. Points $P_1$, $P_2$, and $P_3$ are on $\omega_1$, $\omega_2$, and $\omega_3$ respectively. The lines $P_iP_{i+1}$ are tangent to $\omega_i$ for each $i=1,2,3$, where $P_4 = P_1$. If $P_1P_2 = x$, $P_2P_3 = P_3P_1 = y$ under the condition that $x > y$, find the area of $\triangle P_1P_2P_3$ such that $x + y = 8\sqrt{6}$ (\(x = y + \sqrt{3}y\)). Express the area in the form $\sqrt{a}+\sqrt{b}$ where $a$ and $b$ are positive integers. Calculate the value of $a+b$.

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