(without Phragmén-Lindelöf) $f$ is of exponential type and bounded on the real axis, then there exists $C>0$ such that $|f(x+iy)|\leq Ce^{\tau |y|}$

Question

Question:

Without using Phragmén-Lindelöf, show that if $f$ is of exponential type and uniformly bounded on the real axis, then there exists $C>0$ such that $|f(x+iy)|\leq Ce^{\tau |y|}.$

Definitions:

An entire function $f$ is of exponential type if there exist $K>0$ and $\tau>0$ such that $|f(z)|\leq Ke^{\tau z}.$

Relevant information:

I'm told to consider $f(z)e^{-\tau z}$ for $x\geq 0$ and $f(z)e^{\tau z}$ for $x\leq 0$ and use the maximum principle on appropriate rectangles.

Attempt

I am unsure of how to start this problem.