
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 is defined parametrically by
\[(x,y) = \left( \frac{3 (\sin t - 2)}{3 - \cos t}, \frac{4 (\cos t - 4)}{3 - \cos t} \right).\]
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 $\gcd(|A|,|B|,|C|,|D|,|E|,|F|) = 1.$ Then find $|A| + |B| + |C| + |D| + |E| + |F|.$

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