
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
Two planes $x + y + z = 1$ and $x - 2y + 2z = 4$ intersect along line $L$. Another plane $P$ different from these two planes, contains line $L$ and has a distance of $\frac{3}{\sqrt{14}}$ from the point $(1,2,0)$. Find the equation of plane $P$ in the form 
\[ Ax + By + Cz + D = 0, \]
where $A$, $B$, $C$, and $D$ are integers, $A > 0$, and $\gcd(|A|, |B|, |C|, |D|) = 1.$

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