Example 6 Arrange positive integers starting from 1 according to the pattern shown in Figure 2, where 2 is at the first turn, 3 is at the second turn, 5 is at the third turn, 7 is at the fourth turn, … then the number at the 2007th turn is $\qquad$
To explore the relationship between the number of turns and the numbers at the turns, let $a_{n}$ represent the number corresponding to the $n$-th turn. If we observe in the order of natural numbers, then
$$
\begin{array}{l}
a_{1}=2, a_{2}=3, a_{3}=5, a_{4}=7, a_{5}=10, \\
a_{6}=13, a_{7}=17, a_{8}=21, \cdots \cdots
\end{array}
$$

The relationship between the number of turns $n$ and $a_{n}$ is relatively complex. If we skip some numbers (such as observing in the order of odd numbers), then
$$
a_{3}=a_{1}+3, a_{5}=a_{3}+5, a_{7}=a_{5}+7, \cdots \cdots
$$

That is, the next turning number = the previous turning number + the next turning number.
Thus, $a_{2007}=a_{2005}+2007$
$$
\begin{array}{l}
=a_{2003}+2005+2007=\cdots \\
=a_{3}+5+7+\cdots+2007 \\
=a_{1}+3+5+\cdots+2007 \\
=2+3+5+\cdots+2007 \\
=1+1+3+5+\cdots+2007 \\
=1+\frac{(1+2007) \times 1004}{2}
\end{array}
$$
$$
=1004^{2}+1=1008017 \text {. }
$$

Therefore, the number at the 2007th turn is 1008017.