ISL 2008 A4

For an integer $m$, denote by $t(m)$ the unique number in $\{1,2,3\}$ such that $m+t(m)$ is a multiple of 3. A function $f: \mathbb{Z} \rightarrow \mathbb{Z}$ satisfies $f(-1)=0, f(0)=1, f(1)=-1$ and $$f\left(2^{n}+m\right)=f\left(2^{n}-t(m)\right)-f(m) \text { for all integers } m, n \geq 0 \text { with } 2^{n}>m \text {. }$$ Prove that $f(3 p) \geq 0$ holds for all integers $p \geq 0$.