
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
The ellipse, centered at the origin with one focus at $F_1(0, \sqrt{50})$, intersects the line $y=3x-2$. The midpoint of the chord formed by this intersection has an x-coordinate of $\frac{1}{2}$. Find the equation of the ellipse.

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