
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 integer numbers from  $1$  to  $2002$  are written in a blackboard in increasing order  $1,2,\ldots, 2001,2002$ . After that, somebody erases the numbers in the  $ (3k+1)-th$  places i.e.  $(1,4,7,\dots)$ . After that, the same person erases the numbers in the  $(3k+1)-th$  positions of the new list (in this case,  $2,5,9,\ldots$ ). This process is repeated until one number remains. What is this number?

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