1. Given an arithmetic sequence $\left\{a_{n}\right\}(n=1,2, \cdots)$ with the first term $a$ and common difference $b$, and a geometric sequence $\left\{b_{n}\right\}$ with the first term $b$ and common ratio $a$, where $a$ and $b$ are both positive numbers, and $a_{1}<b_{1}<a_{2}<b_{2}<a_{3}$.
(1) Find the value of $a$;
(2) If for the sequences $\left\{a_{n}\right\}$ and $\left\{b_{n}\right\}$, there exists the relationship $a_{m}+1=b_{n}$, try to find the value of $b$;
(3) For $a_{m}$ that satisfies the relationship in (2), try to find $a_{1}+a_{2}+\cdots+a_{m}$.
(1) $a=2 ;(2) b=3$;
(3) $a_{1}+a_{2}+\cdots+a_{m}=2^{n-2}+3 \times 2^{2 n-3}$.