[数列] 转两道数列问题

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(1)$f(x)=\left[x+\frac12\right],a_1=1,a_{n+1}=a_n+\frac1{a_n}$(n>1),A=$\{m|\left|\{n|f(a_n)=n\}\right|=m\}$,求A
(2)$a_{n+4}=\frac{a_{n+3}+a_{n+2}+a_{n+1}+1}{a_n+1}$收敛于$\sqrt2+1$