
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 alternately choose the sign for each of the numbers 1, 2, ..., 20. Once a sign has been chosen for a number, it cannot be changed. The first player tries to minimize the final absolute value of the total, and the second player tries to maximize it. What is the outcome, assuming both players play perfectly?

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