If $n ( 2 n + 1 ) \int _ { 0 } ^ { 1 } \left( 1 - x ^ { n } \right) ^ { 2 n } d x = 1177 \int _ { 0 } ^ { 1 } \left( 1 - x ^ { n } \right) ^ { 2 n + 1 } d x$, then $n \in N$ is equal to $\_\_\_\_$.
If $n ( 2 n + 1 ) \int _ { 0 } ^ { 1 } \left( 1 - x ^ { n } \right) ^ { 2 n } d x = 1177 \int _ { 0 } ^ { 1 } \left( 1 - x ^ { n } \right) ^ { 2 n + 1 } d x$, then $n \in N$ is equal to $\_\_\_\_$.