已知数列$\left\{a_n\right\}$满足$a_1=1$,$\left|a_{n+1}-a_n\right|=p^n$,$n\in N^{*}$.

(1)若$\left\{a_n\right\}$为递增数列,且$a_1,a_2,a_3$成等差数列,求$P$的值;
(2)若$p=\frac{1}{2}$,且$\left\{a_{2n-1}\right\}$是递增数列,$\left\{a_{2n}\right\}$是递减数列,求数列$\left\{a_n\right\}$的通项公式.