The $8-$queen problem is the problem of putting $8$ chess queens on an $8$ by $8$ chessboard such that none of them is able to capture any other using the standard chess queen$$s moves. The queens must be placed in such a way that no two queens would be able to attack each other.

$1)$ Formulate this problem by defining the initial state, actions, goal test and step cost.

$2)$ Formulate this problem as a Constraint Satisfaction Problem.