Example 5: A total of 240 students participating in a large group performance stand in a row facing the coach, and they report numbers in sequence from left to right as 1, 2, 3, 4, 5, ... The coach asks all students to remember their reported numbers and perform the following actions: First, all students who reported numbers that are multiples of 3 turn around; then, all students who reported numbers that are multiples of 5 turn around; finally, all students who reported numbers that are multiples of 7 turn around. Questions:
(1) How many students are still facing the coach?
(2) Among the students facing the coach, what is the number reported by the 66th student from left to right?
Solution: (1) Let $U=\{1,2, \cdots, 240\}, A_{i}$ represent the set of all multiples of $i$ in $U$. Then
$|U|=240 ;\left|A_{3}\right|=\left[\frac{240}{3}\right]=80$,
$\left|A_{5}\right|=\left[\frac{240}{5}\right]=48,\left|A_{7}\right|=\left[\frac{240}{7}\right]=34$;
$\left|A_{15}\right|=\left[\frac{240}{15}\right]=16,\left|A_{21}\right|=\left[\frac{240}{21}\right]=11$,
$\left|A_{35}\right|=\left[\frac{240}{35}\right]=6 ;\left|A_{105}\right|=\left[\frac{240}{105}\right]=2$.
Therefore, we have $|U|-\left(\left|A_{3}\right|+\left|A_{5}\right|+\left|A_{7}\right|\right)$
$+2\left(\left|A_{15}\right|+\left|A_{21}\right|+\left|A_{35}\right|\right)-4\left|A_{105}\right|=136$ students facing the coach.

If we use a Venn diagram for analysis, fill in the data obtained above into Figure 1, paying attention to the order from inside to outside.

As shown in Figure 1, the students facing the coach are clearly visible, with $109+14+4+9=136$ students.
(2) Using a similar method, we can calculate that among the first 105 students from left to right, 60 students are facing the coach.

Considering that students who report numbers that are not multiples of $3,5,7$ do not turn, they face the coach; students who report numbers that are multiples of two of $3,5,7$ turn twice, they still face the coach; the rest of the students turn once or three times, and they face away from the coach.
Starting from the 106th student, consider the numbers that are multiples of 3, 5, 7.
Multiples of 3 (starting from 105 and adding 3 each time):
$108,111,114,117,120, \cdots$
Multiples of 5 (starting from 105 and adding 5 each time):
$110,115,120, \cdots$
Multiples of 7 (starting from 105 and adding 7 each time):
$112,119, \cdots$
Therefore, starting from the 106th student, the numbers reported by students facing the coach from left to right are:
$$
106,107,109,113,116,118,120, \cdots \text {. }
$$

From this, we can see that the number reported by the 66th student facing the coach is 118.