A binary linear cyclic block code $C_{cyc}(n,k)$ has code length $n=14$ and generator polynomial $g(x)=1+x^3+x^4+x^5.$

$($a$)$ If all the information symbols are $\text{'}1\text{'}$s$,$ what is the corresponding code vector?

$($b$)$ Find the syndrome corresponding to an error in the last information symbol. Is this code capable of correcting this error?