AUSTRIAN-POLISH 2002 Problem 10

For all real number $x$ consider the family $F(x)$ of all sequences $(a_{n})_{n\geq 0}$ satisfying the equation $$a_{n+1}=x-\frac{1}{a_{n}} (n\geq 0).$$ A positive integer $p$ is called a minimal period of the family $F(x)$ if

(a) each sequence $\left(a_{n}\right)\in F(x)$ is periodic with the period $p$ ,

(b) for each $0<q<p$ there exists $\left(a_{n}\right)\in F(x)$ such that $q$ is not a period of $\left(a_{n}\right)$ .

Prove or disprove that for each positive integer $P$ there exists a real number $x=x(P)$ such that the family $F(x)$ has the minimal period $p>P$ .