
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
To pass a test, a thousand wise men are arranged in a column. From hats numbered 1 to 1001, one hat is hidden, and the others are randomly placed on the wise men. Each wise man can only see the numbers on the hats of those standing in front of them. Subsequently, the wise men, starting from the back to the front, call out whole numbers. Each number must be between 1 and 1001, and a number cannot be repeated. The result of the test is the number of wise men who correctly identify the number on their own hat. The wise men knew the conditions of the test in advance and could strategize their actions.

a) Can they guarantee a result greater than 500?

b) Can they guarantee a result of at least 999?

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