Let $a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots$ be a sequence of positive integers in arithmetic progression with common difference 2. Also, let $b _ { 1 } , b _ { 2 } , b _ { 3 } , \ldots$ be a sequence of positive integers in geometric progression with common ratio 2. If $a _ { 1 } = b _ { 1 } = c$, then the number of all possible values of $c$, for which the equality $$2 \left( a _ { 1 } + a _ { 2 } + \cdots + a _ { n } \right) = b _ { 1 } + b _ { 2 } + \cdots + b _ { n }$$ holds for some positive integer $n$, is $\_\_\_\_$
Let $a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots$ be a sequence of positive integers in arithmetic progression with common difference 2. Also, let $b _ { 1 } , b _ { 2 } , b _ { 3 } , \ldots$ be a sequence of positive integers in geometric progression with common ratio 2. If $a _ { 1 } = b _ { 1 } = c$, then the number of all possible values of $c$, for which the equality
$$2 \left( a _ { 1 } + a _ { 2 } + \cdots + a _ { n } \right) = b _ { 1 } + b _ { 2 } + \cdots + b _ { n }$$
holds for some positive integer $n$, is $\_\_\_\_$