3 (Full score: 50 points) Before the World Cup, the coach of country $F$ plans to observe the seven players $A_{1}, A_{2}, \cdots, A_{7}$ by letting 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) of $A_{1}, A_{2}, A_{3}, A_{4}$ is divisible by 7, while the total playing time (in minutes) of $A_{5}, A_{6}, A_{7}$ is divisible 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?
Let the playing time of the $i$-th player be $x_{i}$ minutes $(i=1,2, \cdots, 7)$. The problem is transformed into finding the positive integer solutions of the indeterminate equation
$$
x_{1}+x_{2}+\cdots+x_{7}=270
$$

satisfying the conditions $7 \mid x_{i}(i=1,2,3,4)$, and $13 \mid x_{j}(j=5,6,7)$. Let $x_{1}+x_{2}+x_{3}+x_{4}=7 m, x_{5}+x_{6}+x_{7}=13 n$, then
$$
7 m+13 n=270,
$$

and $m, n \in \mathbf{N}_{+}, m \geqslant 4, n \geqslant 3$.
It is easy to find that the positive integer solutions $(m, n)$ satisfying the above conditions are
$$
(m, n)=(33,3),(20,10),(7,17) .
$$

When $(m, n)=(33,3)$, $x_{5}=x_{6}=x_{7}=13$. Let $x_{i}=7 y_{i}$ $(i=1,2,3,4)$, then
$$
y_{1}+y_{2}+y_{3}+y_{4}=33,
$$

there are $\mathrm{C}_{33-1}^{4}-1=\mathrm{C}_{32}^{3}=4960$ groups of positive integer solutions $\left(y_{1}, y_{2}, y_{3}, y_{4}\right)$, at this time (1) has
4960 groups of positive integer solutions satisfying the conditions.
When $(m, n)=(20,10)$, let $x_{i}=7 y_{i}(i=1,2,3,4)$, $x_{j}=13 y_{j}(j=5,6,7)$, then
$$
y_{1}+y_{2}+y_{3}+y_{4}=20, \quad y_{5}+y_{6}+y_{7}=10,
$$

at this time (1) has $\mathrm{C}_{19}^{3} \times \mathrm{C}_{9}^{2}=34884$ groups of positive integer solutions satisfying the conditions.
When $(m, n)=(7,17)$, let $x_{i}=7 y_{i}(i=1,2,3,4), x_{j}=$ $13 y_{j}(j=5,6,7)$, then
$$
y_{1}+y_{2}+y_{3}+y_{4}=7, \quad y_{5}+y_{6}+y_{7}=17,
$$

at this time (1) has $\mathrm{C}_{6}^{3} \times \mathrm{C}_{16}^{2}=2400$ groups of positive integer solutions satisfying the conditions.
In summary, the number of groups of positive integer solutions of (1) satisfying the conditions is
$$
4960+34884+2400=42244 .
$$