In the game of *Winners Make Zeros*, a pair of positive integers \((m,n)\) is written on a sheet of paper. Then the game begins, as the players make the following legal moves:

- If \(m \geq n\), the player chooses a positive integer \(c\) such that \(m-cn \geq 0\), and replaces \((m,n)\) with \((m-cn,n)\).
- If \(m < n\), the player chooses a positive integer \(c\) such that \(n-cm \geq 0\), and replaces \((m,n)\) with \((m,n-cm)\).

When \(m\) or \(n\) becomes \(0\), the game ends, and the last player to have moved is declared the winner. If \(m\) and \(n\) are originally \(2007777\) and \(2007\), find the largest choice the first player can make for \(c\) (on his first move) such that the first player has a winning strategy after that first move.

Let's think step by step and output the final answer within \boxed{}.
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