RMO 2013 M5

Let $a_1,b_1,c_1$ be natural numbers. We define $$a_2=\gcd(b_1,c_1),\,\,\,\,\,\,\, b_2=\gcd(c_1,a_1),\,\,\,\,\,\,\, c_2=\gcd(a_1,b_1),$$ and $$a_3=\operatorname{lcm}(b_2,c_2),\,\,\,\,\,\,\, b_3=\operatorname{lcm}(c_2,a_2),\,\,\,\,\,\,\, c_3=\operatorname{lcm}(a_2,b_2).$$ Show that $\gcd(b_3,c_3)=a_2$ .