3. Given real numbers $a_{i}(i=1,2, \cdots, 2016)$ satisfy $9 a_{i}>11 a_{i+1}^{2}(i=1,2, \cdots, 2015)$. Find
$$
\begin{aligned}
f= & \left(a_{1}-a_{2}^{2}\right)\left(a_{2}-a_{3}^{2}\right) \cdots\left(a_{2015}-a_{2016}^{2}\right) \cdot \\
& \left(a_{2016}-a_{1}^{2}\right)
\end{aligned}
$$

the maximum value.
(2016, National High School Mathematics Joint Competition)
Hint: Suppose $a_{2016}>a_{1}^{2}$. By the AM-GM inequality and Cauchy-Schwarz inequality, we obtain $f_{\text {max }}=\frac{1}{2^{4033}}$.