For positive integer $x>1$, define set $S_x$ as $$S_x=\{p^\alpha|p \textup{ is one of the prime divisor of }x,\alpha \in \mathbb{N},p^\alpha|x,\alpha \equiv v_p(x)(\textup{mod} 2)\},$$where $v_p(n)$ is the power of prime divisor $p$ in positive integer $n.$ Let $f(x)$ be the sum of all the elements of $S_x$ when $x>1,$ and $f(1)=1.$ Let $m$ be a given positive integer, and the sequence $a_1,a_2,\cdots,a_n,\cdots$ satisfy that for any positive integer $n>m,$ $a_{n+1}=\max\{ f(a_n),f(a_{n-1}+1),\cdots,f(a_{n-m}+m)\}.$ Prove that (1)there exists constant $A,B(0<A<1),$ such that when positive integer $x$ has at least two different prime divisors, $f(x)<Ax+B$ holds; (2)there exists positive integer $Q$, such that for any positive integer $n,a_n<Q.$