
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
Line \( M \) is the intersection of the planes \( 2x - y + 5z = 4 \) and \( x + 4y - 2z = 6 \). There exists a plane \( Q \), which contains line \( M \) and has a perpendicular distance of \( \frac{5}{\sqrt{6}} \) from the point \( (1,2,-2) \). Find the equation of plane \( Q \). The answer should be in the form
\[ Ax + By + Cz + D = 0, \] 
where \( A, B, C, D \) are integers with \( A > 0 \) and \( \gcd(|A|, |B|, |C|, |D|) = 1 \).

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