Let a line \(L\) passing through the point \((1,1,1)\) be perpendicular to both the vectors \(2\overset{^}{i}+2\overset{…

Let a line \(L\) passing through the point \((1,1,1)\) be perpendicular to both the vectors \(2\overset{^}{i}+2\overset{^}{j}+\overset{^}{k}\) and \(\overset{^}{i}+2\overset{^}{j}+2\overset{^}{k}\). If \(P(a,b,c)\) is the foot of perpendicular from the origin on the line \(L\), then the value of \(34(a+b+c)\) is:

[JEE Main 2026, 2 Apr (Shift 1)]