VMO 2013 Problem 2

Define a sequence $\{a_n\}$ as: $\left\{\begin{aligned}& a_1=1 & a_{n+1}=3-\frac{a_{n}+2}{2^{a_{n}}}\ \ \text{for} \ n\geq 1.\end{aligned}\right.$

Prove that this sequence has a finite limit as $n\to+\infty$ . Also determine the limit.