12. (13 points) In a basketball shooting game during a physical education class for senior high school boys, the students play in pairs. In each round of the game, each member of a pair shoots twice, and if the total number of successful shots is no less than 3, the pair is called an "excellent pair." Jia and Yi are in the same group, with the probabilities of Jia and Yi making a shot being $p_{1}$ and $p_{2}$, respectively.
(1) If $p_{1}=\frac{3}{4}, p_{2}=\frac{2}{3}$, find the probability that they will be an excellent pair in the first round of the game.
(2) If $p_{1}+p_{2}=\frac{4}{3}$, and Jia and Yi's pair needs to achieve 16 excellent pair rounds in the game, how many rounds of the game should theoretically be played at a minimum? Also, find the values of $p_{1}$ and $p_{2}$ at this time.
12. (1) Let "they obtain an excellent group in the first round of the game" be event $A$. Then
$$
\begin{array}{l} 
P(A) \\
=\left(\mathrm{C}_{2}^{1} \frac{3}{4} \times \frac{1}{4}\right)\left(\mathrm{C}_{2}^{2} \frac{2}{3} \times \frac{2}{3}\right)+ \\
\left(\mathrm{C}_{2}^{2} \frac{3}{4} \times \frac{3}{4}\right)\left(\mathrm{C}_{2}^{1} \frac{2}{3} \times \frac{1}{3}\right)+ \\
\left(\mathrm{C}_{2}^{2} \frac{3}{4} \times \frac{3}{4}\right)\left(\mathrm{C}_{2}^{2} \frac{2}{3} \times \frac{2}{3}\right) \\
= \frac{2}{3} .
\end{array}
$$
(2) The probability that they obtain an excellent group in the first round of the game is
$$
\begin{aligned}
p= & \mathrm{C}_{2}^{1} p_{1}\left(1-p_{1}\right) \mathrm{C}_{2}^{2} p_{2}^{2}+ \\
& \mathrm{C}_{2}^{2} p_{1}^{2} \mathrm{C}_{2}^{1} p_{2}\left(1-p_{2}\right)+\mathrm{C}_{2}^{2} p_{1}^{2} \mathrm{C}_{2}^{2} p_{2}^{2} \\
= & 2 p_{1} p_{2}\left(p_{1}+p_{2}\right)-3 p_{1}^{2} p_{2}^{2} \\
= & \frac{8}{3} p_{1} p_{2}-3 p_{1}^{2} p_{2}^{2} .
\end{aligned}
$$

Since $0 \leqslant p_{1} \leqslant 1,0 \leqslant p_{2} \leqslant 1$, we have
$\frac{1}{3} \leqslant p_{1} \leqslant 1, \frac{1}{3} \leqslant p_{2} \leqslant 1$.
Also, $p_{1} p_{2} \leqslant\left(\frac{p_{1}+p_{2}}{2}\right)^{2}=\frac{4}{9}$, so
$\frac{1}{9} \leqslant p_{1} p_{2} \leqslant \frac{4}{9}$.
Let $t=p_{1} p_{2}$, then $\frac{1}{9} \leqslant t \leqslant \frac{4}{9}$.
Thus, $p=h(t)=-3 t^{2}+\frac{8}{3} t$.
When $t=\frac{4}{9}$, $p_{\text {max }}=\frac{16}{27}$.
The number of times they obtain an excellent group in $n$ rounds of the game $\xi \sim B(n, p)$.
Given $(n p)_{\text {max }}=16$, we know $n=27$.
Therefore, theoretically, at least 27 rounds of the game are needed, at which time,
$$
p_{1}+p_{2}=\frac{4}{3}, p_{1} p_{2}=\frac{4}{9}, p_{1}=p_{2}=\frac{2}{3} .
$$