Given vectors $\overrightarrow { a _ { 1 } } , \overrightarrow { a _ { 2 } } , \overrightarrow { b _ { 1 } } , \overrightarrow { b _ { 2 } } , \ldots , \overrightarrow { b _ { k } } \left( k \in \mathbb{N} ^ { * } \right)$ that are pairwise non-parallel in the plane, satisfying $\left| \overrightarrow { a _ { 1 } } - \overrightarrow { a _ { 2 } } \right| = 1$ and $\left| \overrightarrow { a _ { i } } - \overrightarrow { b _ { j } } \right| \in \{ 1,2 \}$ (where $i = 1,2$ and $j = 1,2 , \ldots , k$), find the maximum value of $k$ as $\_\_\_\_$