
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
Archimedes planned to count all of the prime numbers between  $2$  and  $1000$  using the Sieve of Eratosthenes as follows:
(a) List the integers from  $2$  to  $1000$ .
(b) Circle the smallest number in the list and call this  $p$ .
(c) Cross out all multiples of  $p$  in the list except for  $p$  itself.
(d) Let  $p$  be the smallest number remaining that is neither circled nor crossed out. Circle  $p$ .
(e) Repeat steps  $(c)$  and  $(d)$  until each number is either circled or crossed out.

At the end of this process, the circled numbers are prime and the crossed out numbers are composite.

Unfortunately, while crossing out the multiples of  $2$ , Archimedes accidentally crossed out two odd primes in addition to crossing out all the even numbers (besides  $2$ ). Otherwise, he executed the algorithm correctly. If the number of circled numbers remaining when Archimedes finished equals the number of primes from  $2$  to  $1000$  (including  $2$ ), then what is the largest possible prime that Archimedes accidentally crossed out?

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