
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 a different parametric representation of an ellipse:
\[
(x, y) = \left( \frac{3(\sin t + 2)}{3 + \cos t}, \frac{4(\cos t - 2)}{3 + \cos t} \right)
\]
Find the equation of the ellipse in the form \[Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0,\] where A, B, C, D, E, and F are integers, and compute \( |A| + |B| + |C| + |D| + |E| + |F| \).

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