
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 center at the origin and one of its foci at $F(0,2)$. The chord formed by the intersection of the ellipse and the line $y=3x+7$ has its midpoint with a y-coordinate of $1$. Determine the equation of this ellipse.

A) $\dfrac{x^{2}}{16}+\dfrac{y^{2}}{20}=1$

B) $\dfrac{x^{2}}{12}+\dfrac{y^{2}}{16}=1$

C) $\dfrac{x^{2}}{12}+\dfrac{y^{2}}{8}=1$

D) $\dfrac{x^{2}}{8}+\dfrac{y^{2}}{12}=1$

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