jmc

We begin by examining the quantity $|\pi -7|$. Since $\pi$ is less than 4, clearly $\pi -7$ will be negative. Hence we must negate this quantity to obtain its absolute value, which is always positive. In other words, $$|\pi -7|=-(\pi -7)=7-\pi .$$Continuing, we next consider the expression $\pi -|\pi -7|$, which reduces to $2\pi -7$ in light of the above computation. Since $\pi$ is less than 3.5, this quantity is also negative. Hence we must negate it just as before when taking absolute value, leading to our final answer of $\boxed{7-2\pi }.$