
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 focus \((4, -2)\) and directrix \(4x + 5y = 20\). Express the equation of the 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