
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
Given the line \(6 x - 5 y - 28 = 0\) and the ellipse \(\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1\) with \(a > b > 0\), and \(a^{2}, b \in \mathbf{Z}\), intersects the ellipse at points \(A\) and \(C\). Let \(B(0, b)\) be the top vertex of the ellipse, and the centroid of \(\triangle ABC\) be the right focus \(F_{2}\) of the ellipse. Find the equation of the ellipse.

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