
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 its foci at \((2, 0)\) and \((8, 0)\). Given that it passes through the point \((5, 3)\), determine the equation of the ellipse in the form
\[
\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1
\]
where \(a, b, h,\) and \(k\) are constants, and \(a\) and \(b\) are positive. Find \(a+k\).

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