
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
Determine the number of decreasing sequences of positive integers \(b_1 \geq b_2 \geq b_3 \geq \cdots \geq b_7 \leq 1500\) such that \(b_i - i\) is divisible by 3 for \(1 \leq i \le 7\). Express the number of such sequences as \({m \choose n}\) for some integers \(m\) and \(n\), and compute the remainder when \(m\) is divided by 1000.

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