In the game of [i]Winners Make Zeros[/i], 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:
[list]
[*] If $m\geq n$, the player choose a positive integer $c$ such that $m-cn\geq 0$, and replaces $(m,n)$ with $(m-cn,n)$.
[*] If $m<n$, the player choose a positive integer $c$ such that $n-cm\geq 0$, and replaces $(m,n)$ with $(m,n-cm)$.
[/list]
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.
To solve this problem, we need to analyze the game and determine the largest choice the first player can make for \( c \) such that the first player has a winning strategy after that first move. We start with the initial pair \((m, n) = (2007777, 2007)\).

1. **Initial Setup and First Move**:
   - Given \( m = 2007777 \) and \( n = 2007 \).
   - The first player can choose \( c \) such that \( m - cn \geq 0 \).
   - We need to find the largest \( c \) such that the first player has a winning strategy.

2. **Reduction to Simpler Problem**:
   - We reduce the problem by considering the pair \((m - cn, n)\).
   - The goal is to find the largest \( c \) such that the resulting pair \((m - cn, n)\) is a winning position for the first player.

3. **Analyzing the Game**:
   - If \( m \geq n \), the player can choose \( c \) such that \( m - cn \geq 0 \).
   - If \( m < n \), the player can choose \( c \) such that \( n - cm \geq 0 \).

4. **Specific Calculation**:
   - We start with \( m = 2007777 \) and \( n = 2007 \).
   - We need to find the largest \( c \) such that \( 2007777 - 2007c \geq 0 \).
   - Solving for \( c \):
     \[
     2007777 - 2007c \geq 0 \implies c \leq \frac{2007777}{2007}
     \]
   - Calculating the value:
     \[
     c \leq \frac{2007777}{2007} \approx 1000.387
     \]
   - Since \( c \) must be an integer, the largest possible \( c \) is \( 1000 \).

5. **Verification of Winning Strategy**:
   - If the first player chooses \( c = 1000 \), the new pair is:
     \[
     (2007777 - 1000 \cdot 2007, 2007) = (777, 2007)
     \]
   - We need to check if \((777, 2007)\) is a winning position for the first player.
   - By the lemma, if \( a \geq 2b \), then \((a, b)\) is a losing position. Here, \( 2007 > 2 \cdot 777 \), so \((777, 2007)\) is a losing position for the first player.
   - Therefore, choosing \( c = 1000 \) is not a winning strategy.

6. **Finding the Correct \( c \)**:
   - We need to find the largest \( c \) such that the resulting position is a winning position for the first player.
   - If the first player chooses \( c = 999 \), the new pair is:
     \[
     (2007777 - 999 \cdot 2007, 2007) = (2784, 2007)
     \]
   - We need to check if \((2784, 2007)\) is a winning position for the first player.
   - Since \( 2784 > 2007 \), the first player can make a move to reduce \( 2784 \) further.

7. **Conclusion**:
   - The largest \( c \) such that the first player has a winning strategy is \( 999 \).

The final answer is \( \boxed{999} \)