
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
Find the greatest positive integer  $n$  such that we can choose  $2007$  different positive integers from  $[2\cdot 10^{n-1},10^{n})$  such that for each two  $1\leq i<j\leq n$  there exists a positive integer  $\overline{a_{1}a_{2}\ldots a_{n}}$  from the chosen integers for which  $a_{j}\geq a_{i}+2$ .

*A. Ivanov, E. Kolev*

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