
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
An ellipse has foci at \((8, 1)\) and \((8, 9)\), and it passes through the point \((17, 5)\). Write the equation of this ellipse in the standard form \[\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1,\] where \(a\), \(b\), \(h\), and \(k\) are constants, with \(a\) and \(b\) positive. Find the ordered quadruple \((a, b, h, k)\).

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