
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 largest positive integer $n$ such that within the interval $\left[2 \times 10^{n-1}, 10^{n}\right)$, it is possible to select 2007 distinct integers, ensuring that for any $i, j (1 \leqslant i < j \leqslant n)$, there exists a selected number $\overline{a_{1} a_{2} \cdots a_{n}}$ for which $a_{i} \geqslant a_{i+2}$.

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