Define$\triangle a_n=a_{n+1}-a_n$ then we get that $\triangle a_n=|\triangle a_{n-1}|-|\triangle a_{n-3}|$ Let $b_n=\max \{ |\triangle a_n|,|\triangle a_{n+1}|,|\triangle a_{n+2}|\}$ then $b_n\ge b_{n+1}\ge \dots$ and if there do not exist $k\in \{1,...14^{14}-2\}$ such that $a_k=a_{k+1}=0$ then $b_n\ge b_{n+2} +1,\forall n\in \{1,...14^{14}-2\}$ then $b_{14^{14}}<0$contradition. so there must exist $a_k=a_{k+1}=0,k\in \{1,...14^{14}-2\}$ then the sequence has the form $0,0,t,t,t,0,t,0,0,t,....$ and$t|a_n,\forall n\in N$so $t=1$ and we know that $a_{7l}\equiv 1(mod 2)$,so $a_{14^{14}}=1$