Compute the exact value of the expression $\left|\pi - | \pi - 7 | \right|$.  Write your answer using only integers and $\pi$, without any absolute value signs.
Since $\pi - 7 < 0$, we have that $|\pi - 7| = 7 - \pi$. So $\left|\pi - | \pi - 7 | \right| = \left|\pi - (7 - \pi)\right|$. This simplifies to $\left|2\pi - 7\right|$. And since $2\pi - 7 < 0$, we have that $\left|2\pi - 7\right| = 7 - (2\pi)$. So $\left|\pi - | \pi - 7 | \right| = 7 - 2\pi$. And that's our final answer.

