1. Try to find a set of natural numbers $x, y, z$ that satisfy the equation $x^{3}+y^{4}=z^{3}$. Is the set of natural number solutions to this equation finite or infinite?
Solution: Since $2^{24}+2^{24}=2^{25}$, then $\left(2^{8}\right)^{3} +\left(2^{0}\right)^{4}=\left(2^{5}\right)^{5}$, which means the three numbers $x=2^{8}=256$, $y=2^{0}=64, z=2^{5}=32$ are solutions to the given equation. Furthermore, it is not difficult to deduce that if $\left(x_{0}, y_{0}, z_{0}\right)$ is a set of natural number solutions to the equation, then $\left(k^{2} x_{0}, k^{15} y_{0}, k^{2} z_{0}\right)$ is also a set of solutions, where $k$ is any natural number. Therefore, the equation has infinitely many solutions in the natural number range.