
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
Points \( A = (8, 17) \) and \( B = (14, 15) \) lie on an ellipse with major radius \(a\) and minor radius \(b\) where \(a = 10\) and \(b = 5\). These points also lie on circle \( \omega \) in the plane. The tangent lines to \( \omega \) at \(A\) and \(B\) intersect at a point on the x-axis. Find the area of \( \omega \).
A) \(\frac{2801\pi}{9}\)
B) \(\frac{2602\pi}{9}\)
C) \(\frac{2555\pi}{9}\)
D) \(\frac{2652\pi}{9}\)
E) \(\frac{2700\pi}{9}\)

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