7. (10 points) In a math competition, there are three problems: $A, B, C$. Among the 39 participants, each person answered at least one question correctly. Among those who answered $A$ correctly, those who only answered $A$ are 5 more than those who answered other questions as well; among those who did not answer $A$ correctly, those who answered $B$ correctly are twice as many as those who answered $C$ correctly; it is also known that the number of people who only answered $A$ correctly is equal to the sum of those who only answered $B$ correctly and those who only answered $C$ correctly. Therefore, the maximum number of people who answered $A$ correctly is $\qquad$.
【Analysis】From the problem, we get the following: The number of people who only answered $A$ correctly is $3 b+a$, and the number of people who answered $A$ and other questions correctly is $3 b+a-5$, so we have: $3 b+a+3 b+a-5+3 b+2 a=39$, simplifying to $4 a+9 b=44$. Then, we need to find the values of $a$ and $b$ by assigning values to $a$ and $b$, and find the maximum values of $a$ and $b$. Since the total number of people who answered $A$ correctly is $3 b+a+3 b+a-5=6 b+2 a-5$, we substitute the maximum values of $a$ and $b$ into $6 b+2 a-5$ to solve the problem.
【Solution】Solution: The number of people who only answered $A$ correctly is $3 b+a$, and the number of people who answered $A$ and other questions correctly is $3 b+a-5$, so we have: $3 b+a+3 b+a-5+3 b+2 a=39$,
simplifying to: $4 a+9 b=44$,
Since $a$ and $b$ are natural numbers, when $a=2$, $b=4$; when $a=11$, $b=0$,
$$
\text { i.e. }\left\{\begin{array} { l } 
{ a = 2 } \\
{ b = 4 }
\end{array} \text { or } \left\{\begin{array}{l}
a=11 \\
b=0
\end{array}\right.\right.
$$

The total number of people who answered $A$ correctly is $3 b+a+3 b+a-5=6 b+2 a-5$. Substituting the maximum values of $a$ and $b$ into $6 b+2 a-5$, we get the maximum number:
$$
\begin{array}{l}
6 \times 4+2 \times 2-5 \\
=24+4-5 \\
=23 \text { (people) }
\end{array}
$$

Answer: The maximum number of people who answered $A$ correctly is 23.
The answer is: 23.