
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
Farmer John is inside an ellipse with reflective sides, described by the equation \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\), with \(a > b > 0\). He is standing at the point \((3, 0)\) and shines a laser pointer in the \(y\)-direction. The light reflects off the ellipse and proceeds directly toward Farmer Brown, traveling a distance of 10 before reaching him. Farmer John then spins around in a circle; wherever he points the laser, the light reflects off the wall and hits Farmer Brown. What is the ordered pair \((a, b)\)?

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