Let
\[z = \frac{(-11 + 13i)^3 \cdot (24 - 7i)^4}{3 + 4i},\]and let $w = \frac{\overline{z}}{z}.$  Compute $|w|.$
Since $|\overline{z}| = |z|$ for any complex number $z,$
\[|w| = \left| \frac{\overline{z}}{z} \right| = \frac{|\overline{z}|}{|z|} = \boxed{1}.\]
1
Level 3
Intermediate Algebra
Problem: Let
\[z = \frac{(-11 + 13i)^3 \cdot (24 - 7i)^4}{3 + 4i},\]and let $w = \frac{\overline{z}}{z}.$  Compute $|w|.$
Solution:<|START_OF_LATENT|><|PRIOR_PREFIX|>To find the magnitude of \( w \), we start by analyzing the expression for \( w \). We have \( w = \frac{\overline{z}}{z} \). The magnitude of a complex number \( z \) is given by \( |z| = \sqrt{z \cdot \overline{z}} \). 

Using this property, we can derive the magnitude of \( w \):
1. Calculate \( |w| = \left| \frac{\overline{z}}{z} \right| = \frac{|\overline{z}|}{|z|} \).
2. The magnitude of the conjugate of a complex number is equal to the magnitude of the number itself, thus \( |\overline{z}| = |z| \).
3. Therefore, we have \( |w| = \frac{|z|}{|z|} = 1 \).

This reasoning shows that the magnitude of \( w \) is independent of the specific values of \( z \) and relies solely on the properties of complex numbers. Hence, the final answer is \( 1 \).<|END_OF_LATENT|> The final answer is $1$.