Let a line \( L_1 \) pass through the origin and be perpendicular to the lines \( L_2:\ \mathbf{\vec r}=(3+t)\mathbf{\hat i}+(2t-1)\mathbf{\hat j}+(2t+4)\mathbf{\hat k} \) and \( L_3:\ \mathbf{\vec r}=(3+2s)\mathbf{\hat i}+(3+2s)\mathbf{\hat j}+(2+s)\mathbf{\hat k},\ t,s\in \mathbb{R} \). If \( (a,b,c),\ a\in \mathbb{Z} \), is the point on \( L_3 \) at a distance of \( \sqrt{17} \) from the point of intersection of \( L_1 \) and \( L_2 \), then \( (a+b+c)^2 \) is equal to \( \_\_\_\_\_ \).
