4. Solve the system

$$
\left\{\begin{array}{c}
\log _{x+a}\left(y^{2}-2 c y+c^{2}\right)+\log _{y-c}\left(x^{2}+2 x a+a^{2}\right)=4 \\
\log _{x+a}(-2 y+3+2 c)+\log _{y-c}(2 x-1+2 a)=2
\end{array}\right.
$$

if $a=8, c=20$. Calculate the values of the expression $x_{k} \cdot y_{k}$ for each solution $\left(x_{k}, y_{k}\right)$ of the system and find the minimum among them. Provide the found minimum value in the answer, rounding it to two decimal places if necessary.
Solution. The first equation of the system for $x>-a, x \neq 1-a, y>c, y \neq 1+c$ is equivalent to the equation

$$
\log _{x+a}(y-c)+\log _{y-c}(x+a)=2 \Longleftrightarrow \log _{x+a}(y-c)=1 \Longleftrightarrow y=x+a+c
$$

or for these values, $y=x+28$. Substituting into the second equation leads to the equation

$$
\log _{x+8}(-2 x-13)+\log _{x+8}(2 x+15)=2
$$

which for $-\frac{15}{2}<x<-\frac{13}{2}, x \neq-7$ is equivalent to the equation

$$
\begin{aligned}
\log _{x+8}(-2 x-13)(2 x+15)= & \log _{x+8}(x+8)^{2} \Longleftrightarrow \\
& \Longleftrightarrow(x+8)^{2}+(2 x+13)(2 x+15)=0 \Longleftrightarrow 5 x^{2}+72 x+259=0
\end{aligned}
$$

Of the two roots $x=-7$ and $x=-\frac{37}{5}$ of the last equation, only the latter falls within the domain of definition. Therefore, $(x ; y)=\left(-\frac{37}{5} ; \frac{103}{5}\right)$, and the desired value is $-\frac{37 \cdot 103}{5 \cdot 5}=-152.44$.

Answer: $-152.44$.