
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
Compute the number of non-decreasing sequences of positive integers $a_1 \le a_2 \le a_3 \le \ldots \le a_{10} \le 2013$, where either $a_i - i$ is even or odd for all $1 \le i \le 10$. Find the remainder of $m$ when $m$ is divided by 1000, where the number of such sequences can be expressed as ${m \choose n}$.

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