Define sequence $(a_n):a_n=2^n+3^n+6^n+1(n\in\mathbb{Z}_+)$. Are there intenger $k\geq2$, satisfying that $\gcd(k,a_i)=1$ for all $k\in\mathbb{Z}_+$? If yes, find the smallest $k$. If not, prove this.
Problem
Source: 2016 China Northern MO, Grade 10, Problem 7; 2016 China Northern MO, Grade 11, Problem 7
Tags: number theory