If the sum of first 11 terms of an A.P., $\mathbf{a}_1, \mathbf{a}_2, \mathbf{a}_3, \ldots$ is $0\left(\mathbf{a}_1 \neq 0\right)$, then the sum of the A.P., $\mathbf{a}_1, \mathbf{a}_3, \mathbf{a}_5, \ldots, \mathbf{a}_{23}$ is $\mathbf{a}_1$, where $\mathbf{k}$ is equal to .
