Suppose that $f$ is entire and that for each $z$, either $|f(z)| \le 1$ or $|f'(z)| \le 1$. How can I prove that $f$ is a linear polynomial?

Question

Suppose that $f$ is entire and that for each $z$, either $|f(z)| \le 1$ or $|f'(z)| \le 1$. How can I prove that $f$ is a linear polynomial?
What I have tried: I tried to use line integral to show $$|f(z)| \le A + |z|$$where $A = max(1, |f(0)|)$
However, I was not satisfied with my approach. I am looking forward to hearing how you tackle it.