
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
Let

\[
N = \sum_{a_1 = 0}^2 \sum_{a_2 = 0}^{a_1} \sum_{a_3 = 0}^{a_2} \dots \sum_{a_{2011} = 0}^{a_{2010}} \left [ \prod_{n=1}^{2011} a_n \right ].
\]

Find the remainder when  $N$  is divided by 1000.

*Proposed by Lewis Chen*

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