Third Question: Before the World Cup, the coach of country $F$ plans to evaluate seven players, $A_{1}, A_{2}, \cdots, A_{7}$, by having them play in three training matches (each 90 minutes long). Assume that at any moment during the matches, exactly one of these players is on the field, and the total playing time (in minutes) for $A_{1}, A_{2}, A_{3}, A_{4}$ can all be divided by 7, while the total playing time (in minutes) for $A_{5}, A_{6}, A_{7}$ can all be divided by 13. If there are no limits on the number of substitutions per match, how many different scenarios are there for the total playing time of each player?
Solution: Let the times for $A_{1}, A_{2}, A_{3}, A_{4}$ be $7 k_{1}, 7 k_{2}$, $7 k_{3}, 7 k_{4}$, and their sum be $x: A_{5}, A_{6}, A_{7}$ have times $13 k_{5}$, $13 k_{6}, 13 k_{7}$, and their sum be $y$.

Then the original problem is to find the number of positive integer solutions $\left(k_{1}, k_{2}, \cdots, k_{7}\right)$ that satisfy $7\left(k_{1}+k_{2}+k_{3}+k_{4}\right)+13\left(k_{5}+\right.$ $\left.k_{6}+k_{7}\right)=3 \times 90=270$. For $7 x+13 y=270, x=\frac{270-13 y}{7} \geqslant 4$ we get
$$
\left\{\begin{array} { l } 
{ x = 3 3 , } \\
{ y = 3 ; }
\end{array} \quad \left\{\begin{array} { l } 
{ x = 2 0 , } \\
{ y = 1 0 ; }
\end{array} \left\{\begin{array}{l}
x=7, \\
y=17 .
\end{array}\right.\right.\right.
$$

Thus, there are only three pairs $(x, y)$ that satisfy the conditions.
Solving the system of equations $\left\{\begin{array}{l}k_{1}+k_{2}+k_{3}+k_{4}=33, \\ k_{5}+k_{6}+k_{7}=3 ;\end{array}\right.$,
$$
\left\{\begin{array} { l } 
{ k _ { 1 } + k _ { 2 } + k _ { 3 } + k _ { 4 } = 2 0 , } \\
{ k _ { 5 } + k _ { 6 } + k _ { 7 } = 1 0 ; }
\end{array} \quad \left\{\begin{array}{l}
k_{1}+k_{2}+k_{3}+k_{4}=7, \\
k_{5}+k_{6}+k_{7}=17 .
\end{array}\right.\right.
$$

There are a total of $\mathrm{C}_{32}^{3} \mathrm{C}_{2}^{2}+\mathrm{C}_{19}^{3} \mathrm{C}_{9}^{2}+\mathrm{C}_{6}^{3} \mathrm{C}_{16}^{2}=42244$ positive integer solutions, and each solution corresponds to a time distribution.
Therefore, there are 42244 cases in total.