
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
Calculate the number of non-decreasing sequences of positive integers $a_1 \le a_2 \le a_3 \le \cdots \le a_{10} \le 3013$ such that each $a_i - i$ is divisible by 3. Express your answer as ${m \choose n}$, where $m > n$, and find the remainder when $m$ is divided by 1000.

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