
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
Three circles with centers P, Q, and R have radii of $\sqrt{2}, 2\sqrt{2},$ and $3\sqrt{2}$, respectively, and are tangent to the same horizontal line $l$. Given that Q' is between P' and R', and each circle is externally tangent to at least one other circle, and circle R is shifted vertically up by $\sqrt{2}$ units compared to P and Q, calculate the area of triangle PQR.

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