Given the following dictionary in Python:

student$\_$data $=\{$

"name": "Alice",

"age": $15,$

"grade": $"$A$-"$

$\}$

Perform the following operations step$-$by$-$step and present your final code:

Check if the key "grade" exists in the student$\_$data dictionary.

Print the value associated with the key "age" in the student$\_$data dictionary.

Add a new key$-$value pair to the student$\_$data dictionary, with the key "city" and the value "New York".

Print the updated student$\_$data dictionary.