
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 an ellipse defined parametrically by:
\[
(x,y) = \left( \frac{4 (\cos t + 2)}{3 + \sin t}, \frac{5 (\sin t - 4)}{3 + \sin t} \right).
\]
Determine 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 $\gcd(|A|,|B|,|C|,|D|,|E|,|F|) = 1.$ Calculate $|A| + |B| + |C| + |D| + |E| + |F|.$

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