已知正项数列\(\{a_{n}\}\)满足\(a_{1}=1\)，\(a_{2}=2\)，且对任意的正整数\(n\)，\(1+a_{n+1}^{2}\)是\(a_{n}^{2}\)和\(a_{n+2}^{2}\)的等差中项．
\((1)\)证明：\(\{a_{n+1}^{2}-a_{n}^{2}\}\)是等差数列，并求\(\{a_{n}\}\)的通项公式；
\((2)\)若\(b_{n}-b_{n-1}=a_{n}\)，且\(b_{1}=a_{1}\)，求数列\(\{b_{n}\}\)的通项公式．