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 $\frac{2}{\sqrt{3}}$ from the point $(3,1,-1).$  Find the equation of plane $P.$  Enter your answer in the form
\[Ax + By + Cz + D = 0,\]where $A,$ $B,$ $C,$ $D$ are integers such that $A > 0$ and $\gcd(|A|,|B|,|C|,|D|) = 1.$
<|im_start|>system
You are a helpful assistant.<|im_end|>
<|im_start|>user
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 $\frac{2}{\sqrt{3}}$ from the point $(3,1,-1).$  Find the equation of plane $P.$  Enter your answer in the form
\[Ax + By + Cz + D = 0,\]where $A,$ $B,$ $C,$ $D$ are integers such that $A > 0$ and $\gcd(|A|,|B|,|C|,|D|) = 1.$
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant

user
5x - 11y + z - 17 = 0
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 $\frac{2}{\sqrt{3}}$ from the point $(3,1,-1).$  Find the equation of plane $P.$  Enter your answer in the form
\[Ax + By + Cz + D = 0,\]where $A,$ $B,$ $C,$ $D$ are integers such that $A > 0$ and $\gcd(|A|,|B|,|C|,|D|) = 1.$
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