A.P + GP $\neq$ A.G.P

Algebra Level 4

Let $A=\{a_1, a_2, \ldots, a_n\}$ be a set of the first $n$ terms of an arithmetic progression. Similarly, let $B=\{b_1, b_2, \ldots, b_n\}$ be a set of the first $n$ terms of a geometric progression.

If a new set $C=A+B=\{a_1+b_1, a_2+b_2, \ldots, a_n+b_n\}$ and the first four terms of $C$ are $\{0, 0, 1, 0\},$ what is the $11^\text{th}$ term of $C?$