
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
Let \(ABC\) be a triangle where \(AB = 9\), \(BC = 10\), and \(CA = 17\). Let \(\Omega\) be its circumcircle, and let \(A_1\), \(B_1\), \(C_1\) be the diametrically opposite points from \(A\), \(B\), and \(C\), respectively, on \(\Omega\). Find the area of the convex hexagon with the vertices \(A, B, C, A_1, B_1, C_1\).

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