Multi-Player Coin Game

Two players are playing a game where \(n\) coins are arranged from left to right in a line with each coin showing heads. On a turn, a player chooses 8 coins in a row such that the leftmost coin shows heads and flips those coins over. The last player who is able to make a move is the winner. Of the \(1031\) different games with \(105 \leq n \leq 1135\), for how many of these does the first player have a winning strategy?

Details and assumptions

The only requirement is that the leftmost coin shows heads. There is no requirement on any of the other 7 coins.

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