
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 $2010$ positive numbers $a_1, a_2, \ldots , a_{2010}$ satisfy
the inequality $a_ia_j \le i+j$ for all distinct indices $i, j$.
Determine, with proof, the largest possible value of the product
$a_1a_2\cdots a_{2010}$.

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