Differential equations — uniqueness and the range of a solution

Consider the following statements:

(P) The initial value problem $y'=f(y)$, where $f(y)=y\cos\frac1y$ for $y\neq0$ and $f(0)=0$, with $y(0)=0$, has at most one solution.

(Q) If $z(x)$ is the solution of the initial value problem $z'=\dfrac{(1-z^4)^{100}}{1+z^2}$, $z(0)=0$, then for each $\alpha\in(0,2)$ there exists $x_{\alpha}\in\mathbb{R}$ such that $z(x_{\alpha})=1-\alpha$.

Then which of the following statements is true?