In $\triangle A B C$, $\angle B>\angle A>\angle C$, and $\angle B, \angle A, \angle C$ form an arithmetic sequence. $I$ is the incenter of $\triangle A B C$, the extension of $B I$ intersects $A C$ at point $D$, and the extension of $C I$ intersects $A B$ at point $E$.
(1) Prove that $B C^{2}=B E \cdot B A+C D \cdot C A$;
(2) Find the value of $D E\left(\frac{1}{B I}+\frac{1}{C I}\right)$.