A number theory problem by Francisco Rodríguez

Given the Fibonacci sequence $\{F_n\}=\{1, 1, 2, 3, 5, 8, \ldots\}$

Find all the couples $\{u,v\}$ of real positive numbers such that

$u(F_n)^2+v(F_{n+1})^2$

Is an element of the Fibonacci sequence

Enter your answer as the sum $u+v$ of all the couples $\{u,v\}$ that satisfies the given condition.