Let $a,b,c$ be poistive integers,and such $\gcd(a,b)=\gcd(b,c)=\gcd(a,c)=1$,fine the all $a,b,c$ such $$a^2+3b^2c^2=7^c$$

I'm not sure that this question has been studied, but I've been trying for a long time$(a,b,c\le 100)$, and there's only one set of solutions：$(a,b,c)=(2,1,1)$,But I can't prove it. I may need your help. Thank you