
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
For the test, a thousand sages are lined up in a column. From the hats numbered 1 to 1001, one is hidden, and the rest are randomly put on the sages. Each sage can only see the numbers on the hats of all those standing in front of them. Then, in order from the back to the front, the sages announce whole numbers out loud. Each number must be between 1 and 1001, and no number can be repeated. The result of the test is the number of sages who state the number of their hat. The sages knew the conditions of the test in advance and could agree on how to act.

a) Can they guarantee a result of more than 500?

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

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