Given a positive integer [tex]m[/tex], how many tuples [tex](x_1, x_2, x_3, \cdots x_{2m})[/tex] are there such that all three of the following conditions are met:

(i) [tex]x_n \in \{-1, 1\}[/tex] for all integers [tex]0\ \textless \ n \leq 2m[/tex]
(ii) [tex]\sum^{2m}_{i=1} x_i=4[/tex]
(iii) [tex]0 \leq \sum^{k}_{i=1} _j\ \textless \ 4[/tex] for all integers [tex]0\ \textless \ k\ \textless \ 2m[/tex]