
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
Consider the ellipse governed by the equation \( 3x^2 + 2xy + 4y^2 - 18x - 28y + 50 = 0 \) in the coordinate plane. Determine the sum of the maximum and minimum values of the ratio \( \frac{y}{x} \) for all points \( (x, y) \) lying on this ellipse.
A) 11
B) 12
C) 13
D) 14
E) 15

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