
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
A parabola has its focus at $(4, -2)$ and the directrix given by the equation $4x + 6y = 24$. Express the equation of this parabola in the form 
\[ ax^2 + bxy + cy^2 + dx + ey + f = 0, \]
where $a, b, c, d, e, f$ are integers, $a$ is a positive integer, and $\gcd(|a|, |b|, |c|, |d|, |e|, |f|) = 1$.

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