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If $x$ is a real number such that $\max(\min(x, 2 - x), x - 4, 2x - 8) = \pi - 3$, then the number of possible values of $x$ is

For all real values of $x$, $\dfrac{3x^2 - 6x + 12}{x^2 + 2x + 4}$ lies between $1$ and $k$, and does not take any value above $k$. Then $k$ equals: