Example 7 Determine the least possible value of the largest term in an arithmetic sequence composed of seven distinct primes. ${ }^{[4]}$
(2005, British Mathematical Olympiad)
【Analysis】Let these seven different prime numbers be $p_{i}(i=1,2$, $\cdots, 7)$, and
$$
\begin{array}{l}
p_{2}=p_{1}+d, p_{3}=p_{1}+2 d, p_{4}=p_{1}+3 d, \\
p_{5}=p_{1}+4 d, p_{6}=p_{1}+5 d, p_{7}=p_{1}+6 d,
\end{array}
$$

where the common difference $d \in \mathbf{Z}_{+}$.
Since $p_{2}=p_{1}+d$ and $p_{3}=p_{1}+2 d$ are two odd primes, the common difference $d$ is even, and $p_{1} \geqslant 3$.

If $3 \nmid d$, then $p_{2} 、 p_{3}=p_{2}+d 、 p_{4}=p_{2}+2 d$ are distinct modulo 3, one of them must be a multiple of 3, which contradicts that $p_{2}$, $p_{3}$, and $p_{4}$ are all primes greater than 3. Hence, $3 \mid d$, and $p_{1} \geqslant 5$.
If $5 \nmid d$, then
$$
\begin{array}{l}
p_{2} 、 p_{3}=p_{2}+d 、 p_{4}=p_{2}+2 d 、 \\
p_{5}=p_{2}+3 d 、 p_{6}=p_{2}+4 d
\end{array}
$$

these five numbers are distinct modulo 5, one of them must be a multiple of 5, which contradicts that they are all primes greater than 5. Therefore, $5 \mid d$, and $p_{1} \geqslant 7$.
In summary, $30 \mid d$, and $p_{1} \geqslant 7$.
If $p_{1}>7$, and $7 \nmid d$, then
$$
\begin{array}{l}
p_{1} 、 p_{2}=p_{1}+d 、 p_{3}=p_{1}+2 d 、 p_{4}=p_{1}+3 d 、 \\
p_{5}=p_{1}+4 d 、 p_{6}=p_{1}+5 d 、 p_{7}=p_{1}+6 d
\end{array}
$$

these seven numbers are distinct modulo 7, one of them must be a multiple of 7, which contradicts that they are all primes greater than 7.
If $p_{1}>7$, and $71 d$, then $2101 d$.
Thus, $p_{7}>7+210 \times 6=1267$.
Next, consider the case where $p_{1}=7, d=30 k$.
Notice that, $187=p_{1}+30 \times 6=11 \times 17$.
Therefore, $k \neq 1,2,3,6$.
Notice that, $247=p_{1}+30 \times 8=13 \times 19$.
Therefore, $k \neq 4$.
When $k=5$, the arithmetic sequence
$$
7,157,307,457,607,757,907
$$

are all primes, and $907<1267$.
In summary, the arithmetic sequence
$$
7,157,307,457,607,757,907
$$

is the solution.
【Note】Enumerating and discussing different cases is a common method for solving number theory problems, and the common difference of an arithmetic sequence is often the subject of such case-by-case analysis.