Four, (Full marks 20 points, (1) 10 points, (2) 10 points) A total of 240 students participating in a large group performance stand in a row facing the coach, and report numbers in sequence from left to right as $1,2,3,4,5, \cdots$. 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 at this point?
(2) Among the students facing the coach, what is the number reported by the 66th student from left to right?
(1) Because all students who report numbers that are multiples of $15$, $21$, and $35$ but not multiples of $105$ turn twice, still facing the coach. Students who report numbers that are multiples of $3$ but not multiples of $15$ or $21$, multiples of $5$ but not multiples of $15$ or $35$, and multiples of $7$ but not multiples of $21$ or $35$ turn once, all facing away from the coach. Students who report numbers that are multiples of $105$ turn three times, all facing away from the coach. According to this, the number of students facing the coach is
$$
\begin{array}{l}
240-\left\{\left[\frac{240}{3}\right]+\left[\frac{240}{5}\right]+\left[\frac{240}{7}\right]\right\}+2\left\{\left[\frac{240}{15}\right]\right. \\
\left.+\left[\frac{240}{21}\right]+\left[\frac{240}{35}\right]\right\}-\cdots\left[\frac{240}{105}\right]=136 .
\end{array}
$$

That is, 136 students are still facing the coach. The coach has 60 students:
$$
\begin{array}{l}
105-\left\{\left[\frac{105}{3}\right]+\left[\frac{105}{5}\right]+\left[\frac{105}{7}\right]\right\} \\
+2\left\{\left[\frac{105}{15}\right]+\left[\frac{105}{21}\right]+\left[\frac{105}{35}\right]\right\}-4\left[\frac{105}{105}\right]=60 .
\end{array}
$$
The numbers reported must be greater than $105$. Now, from the sequence of integers greater than $105$, remove the multiples of $3$, $5$, and $7$ but not multiples of $15$, $21$, and $35$, leaving $106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, \ldots$ Among the remaining numbers, the 6th number is $118$, so the 66th student facing the coach reports the number $118$.