
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
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 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  $(m,n-cm)$ .

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.

Present the answer in LaTex format: \boxed{Your answer}
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