G1.3 Let $P$ and $P+2$ be both prime numbers satisfying $P(P+2) \leq 2007$.
If $S$ represents the sum of such possible values of $P$, find the value of $S$.
$\begin{array}{l}P^{2}+2 P-2007 \leq 0 \\ (P+1)^{2}-2008 \leq 0 \\ (P+1+\sqrt{2008})(P+1-\sqrt{2008}) \leq 0 \\ (P+1+2 \sqrt{502})(P+1-2 \sqrt{502}) \leq 0 \\ -1-2 \sqrt{502} \leq P \leq-1+2 \sqrt{502} \\ P \text { is a prime } \Rightarrow 0<P \leq-1+2 \sqrt{502} \\ 22=\sqrt{484}<\sqrt{502}<\sqrt{529}=23 \\ 43<-1+2 \sqrt{502}<45 \\ \therefore(P, P+2)=(3,5),(5,7),(11,13),(17,19),(29,31),(41,43) \\ S=3+5+11+17+29+41=106\end{array}$