
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
The numbers \(1, 2, 3, \ldots, 400\) are written on 400 cards. Two players, \(A\) and \(B\), play the following game:

1. In the first step, \(A\) takes 200 cards for themselves.
2. \(B\) then takes 100 cards from both the remaining 200 cards and the 200 cards that \(A\) has, totaling 200 cards for themselves, and leaves the remaining 200 cards for \(A\).
3. In the next step, \(A\) takes 100 cards from both players' cards, totaling 200 cards for themselves, leaving the remaining 200 cards for \(B\).

This process continues until \(B\) completes the 200th step. At the end, the sums of the numbers on the cards held by \(A\) and \(B\) are calculated as \(C_A\) and \(C_B\) respectively. Then, player \(A\) pays player \(B\) the difference \(C_B - C_A\).

What is the maximum difference that \(B\) can achieve if both players play optimally?

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