CONO-SUR 2012 Problem 5

$A$ and $B$ play alternating turns on a $2012 \times 2013$ board with enough pieces of the following types:

Type $1$ : Piece like Type $2$ but with one square at the right of the bottom square. Type $2$ : Piece of $2$ consecutive squares, one over another. Type $3$ : Piece of $1$ square.

At his turn, $A$ must put a piece of the type $1$ on available squares of the board. $B$ , at his turn, must put exactly one piece of each type on available squares of the board. The player that cannot do more movements loses. If $A$ starts playing, decide who has a winning strategy.

Note: The pieces can be rotated but cannot overlap; they cannot be out of the board. The pieces of the types $1$ , $2$ and $3$ can be put on exactly $3$ , $2$ and $1$ squares of the board respectively.