A function $f(x) = x^{3} + ax^{2} + bx + 4$ satisfies the following condition for two integers $a$ and $b$. What is the maximum value of $f(1)$? [4 points]\\
For all real numbers $\alpha$, the limit $\lim_{x \rightarrow \alpha} \frac{f(2x+1)}{f(x)}$ exists.