Maths Paper 1, November 2024, question 6.4

The diagram below shows the parabola $f(x)=-x^2+6x+7$ and a straight line $g$. The line $g$ has gradient $2$ and passes through the point $(2;11)$, so $g(x)=2x+7$. The function $k$ is obtained by shifting $f$ horizontally, so that $k(x)=f(x+m)=-x^2-2mx-m^2+6x+6m+7$. Determine the value of $m$ for which $g$ is a tangent to $k$.