
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 $L$ is the intersection of the planes $x + 2y + 3z = 2$ and $x - y + z = 3$. A plane $P$, different from both these planes, contains line $L$, and has a distance of $\frac{3}{\sqrt{6}}$ from the point $(1,0,1)$. Find the equation of plane $P$ such that the equation of $P$ is in the form
\[Ax + By + Cz + D = 0,\]
where $A$, $B$, $C$, and $D$ 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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