
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 $\omega_1$, $\omega_2$, $\omega_3$, and $\omega_4$ each have radius $5$ and are placed in the plane so that each circle is externally tangent to the other two adjacent circles. Points $P_1$, $P_2$, $P_3$, and $P_4$ lie on $\omega_1$, $\omega_2$, $\omega_3$, and $\omega_4$ respectively such that $P_1P_2 = P_2P_3 = P_3P_4 = P_4P_1$ and line $P_iP_{i+1}$ is tangent to $\omega_i$ for each $i=1,2,3,4$ (where $P_5 = P_1$). Calculate the area of $\square P_1P_2P_3P_4$.

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