
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
Lines $L$ is the intersection of the planes $x + y + 2z = 4$ and $2x - y + z = 1.$ A plane $P,$ different from both of these, contains line $L,$ and has a distance of $\frac{5}{\sqrt{6}}$ from the point $(2,2,-2).$ Find the equation of plane $P.$ Express your answer in the form
\[Ax + By + Cz + D = 0,\]where $A,$ $B,$ $C,$ $D$ are integers such that $A > 0$ and $\gcd(|A|,|B|,|C|,|D|) = 1.$

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