Consider a ranked list of $5$ documents for a search query. The relevance grades for each document are as follows:

Relevance grades: $3,2,0,1,3$

$1.$ Calculate the Discounted Cumulative Gain $($DCG$)$ @ $3$ for this ranked list.

$2.$ Also, compute the Idealized DCG $($IDCG$)$ @ $3$ assuming the perfect ranking of

documents.

$3.$ Calculate the Normalized Discounted Cumulative Gain $($NDCG$)$ @ $1$ and NDCG @ $3$ for this scenario.