
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
Two players are playing a game. The first player has 1000 even cards (2, 4, 6, ..., 2000), and the second player has 1001 odd cards (1, 3, 5, ..., 2001). They take turns, starting with the first player. During a turn, the player whose turn it is places one of their cards, and the other player, after seeing it, places one of their cards. The player with the higher card number scores a point, and both cards are discarded. There are 1000 rounds in total (one card of the second player is not used). What is the maximum number of points each player can guarantee for themselves, regardless of how their opponent plays?

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