Two players are playing a game. They start by flipping 7 fair coins and arranging them randomly in a row. On a player's turn, she may flip over one coin from heads to tails, or flip over two adjacent coins which show heads to tails. A player loses if there are only tails remaining and it is now her turn to make a move. For how many of the $$2^7=128$$ possible positions of the coins does the first player win?