Increasing sequence

$\{a_n\}$ is a sequence such that $a_1=1$ , $a_{n+1}=\dfrac{a_n}{a_n+2}$ for positive integer $n \geq 1$ .

$\{b_n\}$ is a sequence such that $b_1=\lambda^2-5\lambda$ , $b_{n+1}=(n-2\lambda)(\dfrac{1}{a_n}+1)$ for positive integer $n \geq 1$ , where $\lambda \in \mathbb R$ .

If $\{b_n\}$ is an increasing sequence for positive integer $n \geq 1$ , what is the range of $\lambda$ ?

Caveat: The recurrence formula for $\{b_n\}$ is not defined at $n=0$ .