
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 real numbers \( a_{1}, a_{2}, \ldots, a_{2010} \) satisfy the inequality \( a_{i} a_{j} \leq i+j \) for all \( 1 \leq i < j \leq 2010 \). Determine, with proof, the largest possible value of the product \( a_{1} a_{2} \ldots a_{2010} \).

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